Introduction to Myself
Rong LuoAssociate Professor of Mathematics
Department of MathematicsWest Virginia University
EducationPh.D. in Mathematics, West Virginia University, 2002M. S. in Computer Science, West Virginia University, 2002M. S. in Mathematics, University of Science and Technology of China,1998B. S. in Mathematics, University of Science and Technology of China, 1996
Rong Luo (WVU) Edge Coloring of Graphs 1 / 105
Introduction to Myself
ExperienceAssociate Professor of Mathematics, West Virginia University, August2012-presentProfessor of Mathematics, Middle Tennessee State University, August2011-July 2012Associate Professor of Mathematics, Middle Tennessee State University,August 2007-July 2011Assistant Professor of Mathematics, Middle Tennessee State University,August 2002-July 2007Research Interests: Graph Theory, Combinatorics, Combinatorial MatrixTheory, Applications of Graph Theory in Chemistry and Biology
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Edge Coloring of Graphs
Rong Luo
Department of MathematicsWest Virginia University
Rong Luo (WVU) Edge Coloring of Graphs 3 / 105
Introduction
In mathematics and computer science, graph theory is the study ofgraphs, which are mathematical structures used to model pairwiserelations between objects.
A graph in this context is made up of vertices or nodes and linescalled edges that connect them.
A graph may be undirected, meaning that there is no distinctionbetween the two vertices associated with each edge.
Directed graph: its edges may be directed from one vertex to another.
Graphs are one of the prime objects of study in discrete mathematics.
If there are two or more edges connecting two vertices, such edges arecalled “parallel edges”.
A graph is simple if it has no parallel edges. Otherwise it is called amultigraph or simply a graph.
Rong Luo (WVU) Edge Coloring of Graphs 4 / 105
Introduction
In mathematics and computer science, graph theory is the study ofgraphs, which are mathematical structures used to model pairwiserelations between objects.
A graph in this context is made up of vertices or nodes and linescalled edges that connect them.
A graph may be undirected, meaning that there is no distinctionbetween the two vertices associated with each edge.
Directed graph: its edges may be directed from one vertex to another.
Graphs are one of the prime objects of study in discrete mathematics.
If there are two or more edges connecting two vertices, such edges arecalled “parallel edges”.
A graph is simple if it has no parallel edges. Otherwise it is called amultigraph or simply a graph.
Rong Luo (WVU) Edge Coloring of Graphs 4 / 105
Introduction
In mathematics and computer science, graph theory is the study ofgraphs, which are mathematical structures used to model pairwiserelations between objects.
A graph in this context is made up of vertices or nodes and linescalled edges that connect them.
A graph may be undirected, meaning that there is no distinctionbetween the two vertices associated with each edge.
Directed graph: its edges may be directed from one vertex to another.
Graphs are one of the prime objects of study in discrete mathematics.
If there are two or more edges connecting two vertices, such edges arecalled “parallel edges”.
A graph is simple if it has no parallel edges. Otherwise it is called amultigraph or simply a graph.
Rong Luo (WVU) Edge Coloring of Graphs 4 / 105
Introduction
In mathematics and computer science, graph theory is the study ofgraphs, which are mathematical structures used to model pairwiserelations between objects.
A graph in this context is made up of vertices or nodes and linescalled edges that connect them.
A graph may be undirected, meaning that there is no distinctionbetween the two vertices associated with each edge.
Directed graph: its edges may be directed from one vertex to another.
Graphs are one of the prime objects of study in discrete mathematics.
If there are two or more edges connecting two vertices, such edges arecalled “parallel edges”.
A graph is simple if it has no parallel edges. Otherwise it is called amultigraph or simply a graph.
Rong Luo (WVU) Edge Coloring of Graphs 4 / 105
Introduction
In mathematics and computer science, graph theory is the study ofgraphs, which are mathematical structures used to model pairwiserelations between objects.
A graph in this context is made up of vertices or nodes and linescalled edges that connect them.
A graph may be undirected, meaning that there is no distinctionbetween the two vertices associated with each edge.
Directed graph: its edges may be directed from one vertex to another.
Graphs are one of the prime objects of study in discrete mathematics.
If there are two or more edges connecting two vertices, such edges arecalled “parallel edges”.
A graph is simple if it has no parallel edges. Otherwise it is called amultigraph or simply a graph.
Rong Luo (WVU) Edge Coloring of Graphs 4 / 105
Introduction
In mathematics and computer science, graph theory is the study ofgraphs, which are mathematical structures used to model pairwiserelations between objects.
A graph in this context is made up of vertices or nodes and linescalled edges that connect them.
A graph may be undirected, meaning that there is no distinctionbetween the two vertices associated with each edge.
Directed graph: its edges may be directed from one vertex to another.
Graphs are one of the prime objects of study in discrete mathematics.
If there are two or more edges connecting two vertices, such edges arecalled “parallel edges”.
A graph is simple if it has no parallel edges. Otherwise it is called amultigraph or simply a graph.
Rong Luo (WVU) Edge Coloring of Graphs 4 / 105
Introduction
In mathematics and computer science, graph theory is the study ofgraphs, which are mathematical structures used to model pairwiserelations between objects.
A graph in this context is made up of vertices or nodes and linescalled edges that connect them.
A graph may be undirected, meaning that there is no distinctionbetween the two vertices associated with each edge.
Directed graph: its edges may be directed from one vertex to another.
Graphs are one of the prime objects of study in discrete mathematics.
If there are two or more edges connecting two vertices, such edges arecalled “parallel edges”.
A graph is simple if it has no parallel edges. Otherwise it is called amultigraph or simply a graph.
Rong Luo (WVU) Edge Coloring of Graphs 4 / 105
Example
Petersen graph
v
The degree of v is 3.The maximum degree ∆ = 3.
Rong Luo (WVU) Edge Coloring of Graphs 5 / 105
Applications in Computer Science
In computer science, graphs are used to represent networks ofcommunication, data organization, computational devices, the flow ofcomputation, etc.
One practical example: The link structure of a website could berepresented by a directed graph.
The vertices are the web pages available at the website and a directededge from page A to page B exists if and only if A contains a link toB.
A similar approach can be taken to problems in travel, biology,computer chip design, and many other fields.
The development of algorithms to handle graphs is therefore of majorinterest in computer science. There, the transformation of graphs isoften formalized and represented by graph rewrite systems.
Rong Luo (WVU) Edge Coloring of Graphs 6 / 105
Applications in Computer Science
In computer science, graphs are used to represent networks ofcommunication, data organization, computational devices, the flow ofcomputation, etc.
One practical example: The link structure of a website could berepresented by a directed graph.
The vertices are the web pages available at the website and a directededge from page A to page B exists if and only if A contains a link toB.
A similar approach can be taken to problems in travel, biology,computer chip design, and many other fields.
The development of algorithms to handle graphs is therefore of majorinterest in computer science. There, the transformation of graphs isoften formalized and represented by graph rewrite systems.
Rong Luo (WVU) Edge Coloring of Graphs 6 / 105
Applications in Computer Science
In computer science, graphs are used to represent networks ofcommunication, data organization, computational devices, the flow ofcomputation, etc.
One practical example: The link structure of a website could berepresented by a directed graph.
The vertices are the web pages available at the website and a directededge from page A to page B exists if and only if A contains a link toB.
A similar approach can be taken to problems in travel, biology,computer chip design, and many other fields.
The development of algorithms to handle graphs is therefore of majorinterest in computer science. There, the transformation of graphs isoften formalized and represented by graph rewrite systems.
Rong Luo (WVU) Edge Coloring of Graphs 6 / 105
Applications in Computer Science
In computer science, graphs are used to represent networks ofcommunication, data organization, computational devices, the flow ofcomputation, etc.
One practical example: The link structure of a website could berepresented by a directed graph.
The vertices are the web pages available at the website and a directededge from page A to page B exists if and only if A contains a link toB.
A similar approach can be taken to problems in travel, biology,computer chip design, and many other fields.
The development of algorithms to handle graphs is therefore of majorinterest in computer science. There, the transformation of graphs isoften formalized and represented by graph rewrite systems.
Rong Luo (WVU) Edge Coloring of Graphs 6 / 105
Applications in Computer Science
In computer science, graphs are used to represent networks ofcommunication, data organization, computational devices, the flow ofcomputation, etc.
One practical example: The link structure of a website could berepresented by a directed graph.
The vertices are the web pages available at the website and a directededge from page A to page B exists if and only if A contains a link toB.
A similar approach can be taken to problems in travel, biology,computer chip design, and many other fields.
The development of algorithms to handle graphs is therefore of majorinterest in computer science. There, the transformation of graphs isoften formalized and represented by graph rewrite systems.
Rong Luo (WVU) Edge Coloring of Graphs 6 / 105
Applications in Physics and Chemistry
Graph theory is also used to study molecules in chemistry and physics.
In chemistry a graph makes a natural model for a molecule, wherevertices represent atoms and edges bonds.
This approach is especially used in computer processing of molecularstructures, ranging from chemical editors to database searching.
Chemical graph theory has been studied extensively.
Rong Luo (WVU) Edge Coloring of Graphs 7 / 105
Applications in Physics and Chemistry
Graph theory is also used to study molecules in chemistry and physics.
In chemistry a graph makes a natural model for a molecule, wherevertices represent atoms and edges bonds.
This approach is especially used in computer processing of molecularstructures, ranging from chemical editors to database searching.
Chemical graph theory has been studied extensively.
Rong Luo (WVU) Edge Coloring of Graphs 7 / 105
Applications in Physics and Chemistry
Graph theory is also used to study molecules in chemistry and physics.
In chemistry a graph makes a natural model for a molecule, wherevertices represent atoms and edges bonds.
This approach is especially used in computer processing of molecularstructures, ranging from chemical editors to database searching.
Chemical graph theory has been studied extensively.
Rong Luo (WVU) Edge Coloring of Graphs 7 / 105
Applications in Physics and Chemistry
Graph theory is also used to study molecules in chemistry and physics.
In chemistry a graph makes a natural model for a molecule, wherevertices represent atoms and edges bonds.
This approach is especially used in computer processing of molecularstructures, ranging from chemical editors to database searching.
Chemical graph theory has been studied extensively.
Rong Luo (WVU) Edge Coloring of Graphs 7 / 105
Applications in Social Science
Graph theory is also widely used in sociology as a way.
Under the umbrella of Social Network graphs there are many differenttypes of graphs:
the Acquaintanceship and Friendship Graphs, these graphs are usefulfor representing whether n people know each other.
The influence graph. This graph is used to model whether certainpeople can influence the behavior of others.
Collaboration graph which models whether two people work togetherin a particular way.
Rong Luo (WVU) Edge Coloring of Graphs 8 / 105
Applications in Social Science
Graph theory is also widely used in sociology as a way.
Under the umbrella of Social Network graphs there are many differenttypes of graphs:
the Acquaintanceship and Friendship Graphs, these graphs are usefulfor representing whether n people know each other.
The influence graph. This graph is used to model whether certainpeople can influence the behavior of others.
Collaboration graph which models whether two people work togetherin a particular way.
Rong Luo (WVU) Edge Coloring of Graphs 8 / 105
Applications in Social Science
Graph theory is also widely used in sociology as a way.
Under the umbrella of Social Network graphs there are many differenttypes of graphs:
the Acquaintanceship and Friendship Graphs, these graphs are usefulfor representing whether n people know each other.
The influence graph. This graph is used to model whether certainpeople can influence the behavior of others.
Collaboration graph which models whether two people work togetherin a particular way.
Rong Luo (WVU) Edge Coloring of Graphs 8 / 105
Applications in Social Science
Graph theory is also widely used in sociology as a way.
Under the umbrella of Social Network graphs there are many differenttypes of graphs:
the Acquaintanceship and Friendship Graphs, these graphs are usefulfor representing whether n people know each other.
The influence graph. This graph is used to model whether certainpeople can influence the behavior of others.
Collaboration graph which models whether two people work togetherin a particular way.
Rong Luo (WVU) Edge Coloring of Graphs 8 / 105
Applications in Social Science
Graph theory is also widely used in sociology as a way.
Under the umbrella of Social Network graphs there are many differenttypes of graphs:
the Acquaintanceship and Friendship Graphs, these graphs are usefulfor representing whether n people know each other.
The influence graph. This graph is used to model whether certainpeople can influence the behavior of others.
Collaboration graph which models whether two people work togetherin a particular way.
Rong Luo (WVU) Edge Coloring of Graphs 8 / 105
Graph Coloring
Graph Coloring is the most active area in graph theory.
In general, a graph coloring problem is to partition the objects of agraph (edges, vertices, faces) under certain rules to minimize thenumber of color classes.
It has many applications in practice.
Vertex coloring: A vertex coloring of a graph is to color the verticesof the graph in such a way that any two adjacent vertices receivedifferent colors.
Edge Coloring: An edge coloring of a graph is to color the edges ofthe graph in such a way that any two adjacent edges receive differentcolors.
Face coloring/Map Coloring
The central problem in coloring problems is to find the minimumnumber of colors needed to color the objects.
Graph Coloring has many applications in job scheduling, assignmentsof classes/classrooms, assignments of wireless channels.
Rong Luo (WVU) Edge Coloring of Graphs 9 / 105
Graph Coloring
Graph Coloring is the most active area in graph theory.
In general, a graph coloring problem is to partition the objects of agraph (edges, vertices, faces) under certain rules to minimize thenumber of color classes.
It has many applications in practice.
Vertex coloring: A vertex coloring of a graph is to color the verticesof the graph in such a way that any two adjacent vertices receivedifferent colors.
Edge Coloring: An edge coloring of a graph is to color the edges ofthe graph in such a way that any two adjacent edges receive differentcolors.
Face coloring/Map Coloring
The central problem in coloring problems is to find the minimumnumber of colors needed to color the objects.
Graph Coloring has many applications in job scheduling, assignmentsof classes/classrooms, assignments of wireless channels.
Rong Luo (WVU) Edge Coloring of Graphs 9 / 105
Graph Coloring
Graph Coloring is the most active area in graph theory.
In general, a graph coloring problem is to partition the objects of agraph (edges, vertices, faces) under certain rules to minimize thenumber of color classes.
It has many applications in practice.
Vertex coloring: A vertex coloring of a graph is to color the verticesof the graph in such a way that any two adjacent vertices receivedifferent colors.
Edge Coloring: An edge coloring of a graph is to color the edges ofthe graph in such a way that any two adjacent edges receive differentcolors.
Face coloring/Map Coloring
The central problem in coloring problems is to find the minimumnumber of colors needed to color the objects.
Graph Coloring has many applications in job scheduling, assignmentsof classes/classrooms, assignments of wireless channels.
Rong Luo (WVU) Edge Coloring of Graphs 9 / 105
Graph Coloring
Graph Coloring is the most active area in graph theory.
In general, a graph coloring problem is to partition the objects of agraph (edges, vertices, faces) under certain rules to minimize thenumber of color classes.
It has many applications in practice.
Vertex coloring: A vertex coloring of a graph is to color the verticesof the graph in such a way that any two adjacent vertices receivedifferent colors.
Edge Coloring: An edge coloring of a graph is to color the edges ofthe graph in such a way that any two adjacent edges receive differentcolors.
Face coloring/Map Coloring
The central problem in coloring problems is to find the minimumnumber of colors needed to color the objects.
Graph Coloring has many applications in job scheduling, assignmentsof classes/classrooms, assignments of wireless channels.
Rong Luo (WVU) Edge Coloring of Graphs 9 / 105
Graph Coloring
Graph Coloring is the most active area in graph theory.
In general, a graph coloring problem is to partition the objects of agraph (edges, vertices, faces) under certain rules to minimize thenumber of color classes.
It has many applications in practice.
Vertex coloring: A vertex coloring of a graph is to color the verticesof the graph in such a way that any two adjacent vertices receivedifferent colors.
Edge Coloring: An edge coloring of a graph is to color the edges ofthe graph in such a way that any two adjacent edges receive differentcolors.
Face coloring/Map Coloring
The central problem in coloring problems is to find the minimumnumber of colors needed to color the objects.
Graph Coloring has many applications in job scheduling, assignmentsof classes/classrooms, assignments of wireless channels.
Rong Luo (WVU) Edge Coloring of Graphs 9 / 105
Graph Coloring
Graph Coloring is the most active area in graph theory.
In general, a graph coloring problem is to partition the objects of agraph (edges, vertices, faces) under certain rules to minimize thenumber of color classes.
It has many applications in practice.
Vertex coloring: A vertex coloring of a graph is to color the verticesof the graph in such a way that any two adjacent vertices receivedifferent colors.
Edge Coloring: An edge coloring of a graph is to color the edges ofthe graph in such a way that any two adjacent edges receive differentcolors.
Face coloring/Map Coloring
The central problem in coloring problems is to find the minimumnumber of colors needed to color the objects.
Graph Coloring has many applications in job scheduling, assignmentsof classes/classrooms, assignments of wireless channels.
Rong Luo (WVU) Edge Coloring of Graphs 9 / 105
Graph Coloring
Graph Coloring is the most active area in graph theory.
In general, a graph coloring problem is to partition the objects of agraph (edges, vertices, faces) under certain rules to minimize thenumber of color classes.
It has many applications in practice.
Vertex coloring: A vertex coloring of a graph is to color the verticesof the graph in such a way that any two adjacent vertices receivedifferent colors.
Edge Coloring: An edge coloring of a graph is to color the edges ofthe graph in such a way that any two adjacent edges receive differentcolors.
Face coloring/Map Coloring
The central problem in coloring problems is to find the minimumnumber of colors needed to color the objects.
Graph Coloring has many applications in job scheduling, assignmentsof classes/classrooms, assignments of wireless channels.
Rong Luo (WVU) Edge Coloring of Graphs 9 / 105
Graph Coloring
Graph Coloring is the most active area in graph theory.
In general, a graph coloring problem is to partition the objects of agraph (edges, vertices, faces) under certain rules to minimize thenumber of color classes.
It has many applications in practice.
Vertex coloring: A vertex coloring of a graph is to color the verticesof the graph in such a way that any two adjacent vertices receivedifferent colors.
Edge Coloring: An edge coloring of a graph is to color the edges ofthe graph in such a way that any two adjacent edges receive differentcolors.
Face coloring/Map Coloring
The central problem in coloring problems is to find the minimumnumber of colors needed to color the objects.
Graph Coloring has many applications in job scheduling, assignmentsof classes/classrooms, assignments of wireless channels.
Rong Luo (WVU) Edge Coloring of Graphs 9 / 105
Example
Petersen graph
v
The degree of v is 3.The maximum degree ∆ = 3.
Rong Luo (WVU) Edge Coloring of Graphs 10 / 105
Example-Vertex Coloring-3-cloring of the Peterson Graph
Rong Luo (WVU) Edge Coloring of Graphs 11 / 105
Example–Edge Coloring
4−edge coloring of the petersen graph
v
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An Edge Coloring Application
Suppose transceivers u and w send a message to v . They must usedistinct frequencies. Otherwise, v will not be able to understand theirmessage as they will interfere with each other.
Suppose that transceivers u wants to communicate with transceiverv , transceiver w wants to communicate with transceiver x , and v andw are close.
If u and w send messages on a same frequency, v will receive bothmessages on the same frequency and so messages will interfere witheach color.
Solving the frequency assignment problem is equivalent to finding anedge coloring satisfying: (1) adjacent edges are colored with differentcolors and (2) two edges adjacent to a common edge are coloreddifferently.
Rong Luo (WVU) Edge Coloring of Graphs 17 / 105
An Edge Coloring Application
Suppose transceivers u and w send a message to v . They must usedistinct frequencies. Otherwise, v will not be able to understand theirmessage as they will interfere with each other.
Suppose that transceivers u wants to communicate with transceiverv , transceiver w wants to communicate with transceiver x , and v andw are close.
If u and w send messages on a same frequency, v will receive bothmessages on the same frequency and so messages will interfere witheach color.
Solving the frequency assignment problem is equivalent to finding anedge coloring satisfying: (1) adjacent edges are colored with differentcolors and (2) two edges adjacent to a common edge are coloreddifferently.
Rong Luo (WVU) Edge Coloring of Graphs 17 / 105
An Edge Coloring Application
Suppose transceivers u and w send a message to v . They must usedistinct frequencies. Otherwise, v will not be able to understand theirmessage as they will interfere with each other.
Suppose that transceivers u wants to communicate with transceiverv , transceiver w wants to communicate with transceiver x , and v andw are close.
If u and w send messages on a same frequency, v will receive bothmessages on the same frequency and so messages will interfere witheach color.
Solving the frequency assignment problem is equivalent to finding anedge coloring satisfying: (1) adjacent edges are colored with differentcolors and (2) two edges adjacent to a common edge are coloreddifferently.
Rong Luo (WVU) Edge Coloring of Graphs 17 / 105
An Edge Coloring Application
Suppose transceivers u and w send a message to v . They must usedistinct frequencies. Otherwise, v will not be able to understand theirmessage as they will interfere with each other.
Suppose that transceivers u wants to communicate with transceiverv , transceiver w wants to communicate with transceiver x , and v andw are close.
If u and w send messages on a same frequency, v will receive bothmessages on the same frequency and so messages will interfere witheach color.
Solving the frequency assignment problem is equivalent to finding anedge coloring satisfying: (1) adjacent edges are colored with differentcolors and (2) two edges adjacent to a common edge are coloreddifferently.
Rong Luo (WVU) Edge Coloring of Graphs 17 / 105
Introduction
Tait first studied edge coloring problem in 1880 when he tried toprove the Four Color Conjecture.
He proved that the Four Color Conjecture is equivalent to that every2-edge connected planar cubic graph is edge 3-colorable.
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Introduction
Tait first studied edge coloring problem in 1880 when he tried toprove the Four Color Conjecture.
He proved that the Four Color Conjecture is equivalent to that every2-edge connected planar cubic graph is edge 3-colorable.
Rong Luo (WVU) Edge Coloring of Graphs 19 / 105
Four Color Theorem
Theorem
(Four Color Theorem, Appel and Haken, 1976) Every map can be coloredwith four colors such that any two neighboring regions are colored withdifferent colors.
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History of the Four Color Theorem
The Four Colour Conjecture was first made by Francis Guthrie inOctober 1852 while trying to color the map of counties of England henoticed that four colors sufficed.
He asked his brother Frederick if it was true that any map can becolored using four colors in such a way that adjacent regions receivedifferent colors.
Frederick Guthrie then communicated the conjecture to De Morgan.
The first printed reference is due to Cayley in 1878.
Appel and Haken proved the conjecture in 1976.
Rong Luo (WVU) Edge Coloring of Graphs 22 / 105
History of the Four Color Theorem
The Four Colour Conjecture was first made by Francis Guthrie inOctober 1852 while trying to color the map of counties of England henoticed that four colors sufficed.
He asked his brother Frederick if it was true that any map can becolored using four colors in such a way that adjacent regions receivedifferent colors.
Frederick Guthrie then communicated the conjecture to De Morgan.
The first printed reference is due to Cayley in 1878.
Appel and Haken proved the conjecture in 1976.
Rong Luo (WVU) Edge Coloring of Graphs 22 / 105
History of the Four Color Theorem
The Four Colour Conjecture was first made by Francis Guthrie inOctober 1852 while trying to color the map of counties of England henoticed that four colors sufficed.
He asked his brother Frederick if it was true that any map can becolored using four colors in such a way that adjacent regions receivedifferent colors.
Frederick Guthrie then communicated the conjecture to De Morgan.
The first printed reference is due to Cayley in 1878.
Appel and Haken proved the conjecture in 1976.
Rong Luo (WVU) Edge Coloring of Graphs 22 / 105
History of the Four Color Theorem
The Four Colour Conjecture was first made by Francis Guthrie inOctober 1852 while trying to color the map of counties of England henoticed that four colors sufficed.
He asked his brother Frederick if it was true that any map can becolored using four colors in such a way that adjacent regions receivedifferent colors.
Frederick Guthrie then communicated the conjecture to De Morgan.
The first printed reference is due to Cayley in 1878.
Appel and Haken proved the conjecture in 1976.
Rong Luo (WVU) Edge Coloring of Graphs 22 / 105
History of the Four Color Theorem
The Four Colour Conjecture was first made by Francis Guthrie inOctober 1852 while trying to color the map of counties of England henoticed that four colors sufficed.
He asked his brother Frederick if it was true that any map can becolored using four colors in such a way that adjacent regions receivedifferent colors.
Frederick Guthrie then communicated the conjecture to De Morgan.
The first printed reference is due to Cayley in 1878.
Appel and Haken proved the conjecture in 1976.
Rong Luo (WVU) Edge Coloring of Graphs 22 / 105
Two fallacious proofs of the Four Color Theorem– Kempe
Kempe “proved” it in 1879.
Found to be flawed by Heawood in 1890.
Heawood proved that 5 colors are enough.
Kempe’s error proved very difficult to patch up, but in fact theeventual solution used important ideas that can be traced back toKempe.
Rong Luo (WVU) Edge Coloring of Graphs 23 / 105
Two fallacious proofs of the Four Color Theorem– Kempe
Kempe “proved” it in 1879.
Found to be flawed by Heawood in 1890.
Heawood proved that 5 colors are enough.
Kempe’s error proved very difficult to patch up, but in fact theeventual solution used important ideas that can be traced back toKempe.
Rong Luo (WVU) Edge Coloring of Graphs 23 / 105
Two fallacious proofs of the Four Color Theorem– Kempe
Kempe “proved” it in 1879.
Found to be flawed by Heawood in 1890.
Heawood proved that 5 colors are enough.
Kempe’s error proved very difficult to patch up, but in fact theeventual solution used important ideas that can be traced back toKempe.
Rong Luo (WVU) Edge Coloring of Graphs 23 / 105
Two fallacious proofs of the Four Color Theorem– Kempe
Kempe “proved” it in 1879.
Found to be flawed by Heawood in 1890.
Heawood proved that 5 colors are enough.
Kempe’s error proved very difficult to patch up, but in fact theeventual solution used important ideas that can be traced back toKempe.
Rong Luo (WVU) Edge Coloring of Graphs 23 / 105
Two fallacious proofs of the Four Color Theorem– Tait
Tait “proved” it in 1880.
Found an equivalent formulation of the 4CT in terms of three-edgecoloring.
Found to be flawed by Petersen in 1891.
His results stimulated interest in edge-coloring.
Rong Luo (WVU) Edge Coloring of Graphs 24 / 105
Two fallacious proofs of the Four Color Theorem– Tait
Tait “proved” it in 1880.
Found an equivalent formulation of the 4CT in terms of three-edgecoloring.
Found to be flawed by Petersen in 1891.
His results stimulated interest in edge-coloring.
Rong Luo (WVU) Edge Coloring of Graphs 24 / 105
Two fallacious proofs of the Four Color Theorem– Tait
Tait “proved” it in 1880.
Found an equivalent formulation of the 4CT in terms of three-edgecoloring.
Found to be flawed by Petersen in 1891.
His results stimulated interest in edge-coloring.
Rong Luo (WVU) Edge Coloring of Graphs 24 / 105
Two fallacious proofs of the Four Color Theorem– Tait
Tait “proved” it in 1880.
Found an equivalent formulation of the 4CT in terms of three-edgecoloring.
Found to be flawed by Petersen in 1891.
His results stimulated interest in edge-coloring.
Rong Luo (WVU) Edge Coloring of Graphs 24 / 105
Cayley’s Reduction to Cubic Map
• Cayley observed that it is sufficient to prove that any cubic map can becolored with four colors.• A cubic map is a map in which there are exactly three countries at eachmeeting point.
Rong Luo (WVU) Edge Coloring of Graphs 25 / 105
Cayley’s Reduction to Cubic Map
• Cayley observed that it is sufficient to prove that any cubic map can becolored with four colors.• A cubic map is a map in which there are exactly three countries at eachmeeting point.
Orginal Map
Rong Luo (WVU) Edge Coloring of Graphs 26 / 105
Cayley’s Reduction to Cubic Map
• Cayley observed that it is sufficient to prove that any cubic map can becolored with four colors.• A cubic map is a map in which there are exactly three countries at eachmeeting point.
Add A patchOrginal Map
Rong Luo (WVU) Edge Coloring of Graphs 27 / 105
Cayley’s Reduction to Cubic Map
• Cayley observed that it is sufficient to prove that any cubic map can becolored with four colors.• A cubic map is a map in which there are exactly three countries at eachmeeting point.
Color the new map with 4 colors
A
B
B
D
AC
Rong Luo (WVU) Edge Coloring of Graphs 28 / 105
Cayley’s Reduction to Cubic Map
• Cayley observed that it is sufficient to prove that any cubic map can becolored with four colors.• A cubic map is a map in which there are exactly three countries at eachmeeting point.
D
A
B
B
AC
A
B
CA
B
Remove the patch Color the new map with 4 colors
Rong Luo (WVU) Edge Coloring of Graphs 29 / 105
Cayley’s Reduction to Cubic Map
Lemma
(Cayley) Every plane map is face 4-colorable ⇐⇒ Every cubic plane mapis face 4-colorable.
Rong Luo (WVU) Edge Coloring of Graphs 30 / 105
Tait’s Theorem
Lemma
(Cayley) Every plane map is face 4-colorable ⇐⇒ Every cubic plane mapis face 4-colorable.
Theorem
(Tait, 1880) Every cubic plane map is face 4-colorable ⇐⇒ Every cubicplane map is edge 3-colorable.
Theorem
The four color theorem is equivalent to that every 2-edge connected cubicplane graph is edge 3-colorable.
Rong Luo (WVU) Edge Coloring of Graphs 31 / 105
Tait’s Theorem
Lemma
(Cayley) Every plane map is face 4-colorable ⇐⇒ Every cubic plane mapis face 4-colorable.
Theorem
(Tait, 1880) Every cubic plane map is face 4-colorable ⇐⇒ Every cubicplane map is edge 3-colorable.
Theorem
The four color theorem is equivalent to that every 2-edge connected cubicplane graph is edge 3-colorable.
Rong Luo (WVU) Edge Coloring of Graphs 31 / 105
Tait’s Theorem
Lemma
(Cayley) Every plane map is face 4-colorable ⇐⇒ Every cubic plane mapis face 4-colorable.
Theorem
(Tait, 1880) Every cubic plane map is face 4-colorable ⇐⇒ Every cubicplane map is edge 3-colorable.
Theorem
The four color theorem is equivalent to that every 2-edge connected cubicplane graph is edge 3-colorable.
Rong Luo (WVU) Edge Coloring of Graphs 31 / 105
Face 4-coloring and Edge 3-coloring
Theorem
(Tait, 1880) Every cubic plane map is face 4-colorable ⇐⇒ Every cubicplane map is edge 3-colorable.
Edge 3−coloring
A
B
C
B
D
D
Face 4−coloring
Rong Luo (WVU) Edge Coloring of Graphs 32 / 105
From face 4-coloring to edge 3-coloring
D
A
B
C
B
DA
B
DC
B
D
{AB, CD} = RED
Face 4−coloring to Edge 3−coloring
Rong Luo (WVU) Edge Coloring of Graphs 33 / 105
From face 4-coloring to edge 3-coloring
D
A
B
C
B
DA
B
DC
B
D
{AB, CD} = RED
Face 4−coloring to Edge 3−coloring
Rong Luo (WVU) Edge Coloring of Graphs 34 / 105
From face 4-coloring to edge 3-coloring
{AC,BD} = BLUE
A
B
C
B
DA
B
DC
B
D
{AB, CD} = RED
Face 4−coloring to Edge 3−coloring
D
Rong Luo (WVU) Edge Coloring of Graphs 35 / 105
From face 4-coloring to edge 3-coloring
CA
B
B
DA
B
DC
B
D
{AB, CD} = RED
Face 4−coloring to Edge 3−coloring
D
{AC,BD} = BLUE
Rong Luo (WVU) Edge Coloring of Graphs 36 / 105
From face 4-coloring to edge 3-coloring
{AD,BC} = GREEN
A
B
C
B
DA
B
DC
B
D
{AB, CD} = RED
Face 4−coloring to Edge 3−coloring
D
{AC,BD} = BLUE
Rong Luo (WVU) Edge Coloring of Graphs 37 / 105
From face 4-coloring to edge 3-coloring
{AD,BC} = GREEN
A
B
C
B
DA
B
DC
B
D
{AB, CD} = RED
Face 4−coloring to Edge 3−coloring
D
{AC,BD} = BLUE
Rong Luo (WVU) Edge Coloring of Graphs 38 / 105
From face 4-coloring to edge 3-coloring
• The edges are colored with THREE colors: Red, Blue, Green• Two edges sharing a common endvertex are colored with differentcolors(?)
y
x
u
Rong Luo (WVU) Edge Coloring of Graphs 39 / 105
From face 4-coloring to edge 3-coloring
• The edges are colored with THREE colors: Red, Blue, Green• Two edges sharing a common endvertex are colored with differentcolors(?)
zx
u
y
Rong Luo (WVU) Edge Coloring of Graphs 40 / 105
From face 4-coloring to edge 3-coloring
• The edges are colored with THREE colors: Red, Blue, Green• Two edges sharing a common endvertex are colored with differentcolors(?)
{AD,BC} = GREEN
x
u
y
z
A B
C
{AB, CD} = RED {AC,BD} = BLUE
Rong Luo (WVU) Edge Coloring of Graphs 41 / 105
From face 4-coloring to edge 3-coloring
• The edges are colored with THREE colors: Red, Blue, Green• Two edges sharing a common endvertex are colored with differentcolors(?)
{AD,BC} = GREEN
x
u
y
z
A B
C
{AB, CD} = RED {AC,BD} = BLUE
• If a cubic plane graph has a face 4-coloring, then it has an edge3-coloring.
Rong Luo (WVU) Edge Coloring of Graphs 42 / 105
From edge 3-coloring to face 4-coloring
Face 4−coloring to Edge 3−coloring
A
B
C
B
DA
B
DC
B
D
D
{AB, CD} = RED {AC, BD} = BLUE {AD, BC} = GREEN
Rong Luo (WVU) Edge Coloring of Graphs 43 / 105
From edge 3-coloring to face 4-coloring
D
{AB, CD} = RED {AC, BD} = BLUE {AD, BC} = GREEN
Edge 3−coloring to Face 4−coloring
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From edge 3-coloring to face 4-coloring
B
{AB, CD} = RED {AC, BD} = BLUE {AD, BC} = GREEN
Edge 3−coloring to Face 4−coloring
D
Rong Luo (WVU) Edge Coloring of Graphs 45 / 105
From edge 3-coloring to face 4-coloring
D
{AB, CD} = RED {AC, BD} = BLUE {AD, BC} = GREEN
Edge 3−coloring to Face 4−coloring
D
B
B
Rong Luo (WVU) Edge Coloring of Graphs 46 / 105
From edge 3-coloring to face 4-coloring
D
{AB, CD} = RED {AC, BD} = BLUE {AD, BC} = GREEN
Edge 3−coloring to Face 4−coloring
D
B
B
Rong Luo (WVU) Edge Coloring of Graphs 47 / 105
From edge 3-coloring to face 4-coloring
C
{AB, CD} = RED {AC, BD} = BLUE {AD, BC} = GREEN
Edge 3−coloring to Face 4−coloring
D
B
B
D
A
Rong Luo (WVU) Edge Coloring of Graphs 48 / 105
Introduction– Vizing’s Theorem
The basic question: Given a graph G , what is the smallest number ofcolors needed to color the edges?
This number, denoted χe(G ), is called the edge chromatic number ofG .
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Introduction– Vizing’s Theorem
The basic question: Given a graph G , what is the smallest number ofcolors needed to color the edges?
This number, denoted χe(G ), is called the edge chromatic number ofG .
Rong Luo (WVU) Edge Coloring of Graphs 49 / 105
A natural lower bound
At least ∆ colors are needed. So χe(G ) ≥ ∆. How big could χe(G ) be?
Rong Luo (WVU) Edge Coloring of Graphs 53 / 105
Introduction– Vizing’s Theorem
In 1964, a breakthrough came.
Theorem
(Vizing’s Theorem) For each simple graph G , χe(G ) = ∆ or ∆ + 1.
Classification of graphs:
Class one: χe(G ) = ∆
Class two: χe(G ) = ∆ + 1
Theorem
(Holyer, 1980) The problem of determining whether a graph is class one orclass two is NP-hard.
Rong Luo (WVU) Edge Coloring of Graphs 54 / 105
Introduction– Vizing’s Theorem
In 1964, a breakthrough came.
Theorem
(Vizing’s Theorem) For each simple graph G , χe(G ) = ∆ or ∆ + 1.
Classification of graphs:
Class one: χe(G ) = ∆
Class two: χe(G ) = ∆ + 1
Theorem
(Holyer, 1980) The problem of determining whether a graph is class one orclass two is NP-hard.
Rong Luo (WVU) Edge Coloring of Graphs 54 / 105
Introduction– Vizing’s Theorem
In 1964, a breakthrough came.
Theorem
(Vizing’s Theorem) For each simple graph G , χe(G ) = ∆ or ∆ + 1.
Classification of graphs:
Class one: χe(G ) = ∆
Class two: χe(G ) = ∆ + 1
Theorem
(Holyer, 1980) The problem of determining whether a graph is class one orclass two is NP-hard.
Rong Luo (WVU) Edge Coloring of Graphs 54 / 105
Introduction– Vizing’s Theorem
In 1964, a breakthrough came.
Theorem
(Vizing’s Theorem) For each simple graph G , χe(G ) = ∆ or ∆ + 1.
Classification of graphs:
Class one: χe(G ) = ∆
Class two: χe(G ) = ∆ + 1
Theorem
(Holyer, 1980) The problem of determining whether a graph is class one orclass two is NP-hard.
Rong Luo (WVU) Edge Coloring of Graphs 54 / 105
Introduction– Vizing’s Theorem
In 1964, a breakthrough came.
Theorem
(Vizing’s Theorem) For each simple graph G , χe(G ) = ∆ or ∆ + 1.
Classification of graphs:
Class one: χe(G ) = ∆
Class two: χe(G ) = ∆ + 1
Theorem
(Holyer, 1980) The problem of determining whether a graph is class one orclass two is NP-hard.
Rong Luo (WVU) Edge Coloring of Graphs 54 / 105
Introduction– Vizing’s Theorem
In 1964, a breakthrough came.
Theorem
(Vizing’s Theorem) For each simple graph G , χe(G ) = ∆ or ∆ + 1.
Classification of graphs:
Class one: χe(G ) = ∆
Class two: χe(G ) = ∆ + 1
Theorem
(Holyer, 1980) The problem of determining whether a graph is class one orclass two is NP-hard.
Rong Luo (WVU) Edge Coloring of Graphs 54 / 105
The Petersen graph is class two
4−edge coloring of the petersen graph
v
Rong Luo (WVU) Edge Coloring of Graphs 63 / 105
The Petersen graph is class two
Petersen graph
v
Suppose that we can color the edges of the Petersen graph with threecolors: red, blue, and green.
Rong Luo (WVU) Edge Coloring of Graphs 64 / 105
The Petersen graph is class two
The outer cycle sees all three colors.
The outer 5−cycle is colored with three colors
v
Rong Luo (WVU) Edge Coloring of Graphs 65 / 105
The Petersen graph is class two
Each vertex sees all 3 colors.
Each vertex sees all three colors
v
Rong Luo (WVU) Edge Coloring of Graphs 66 / 105
The Petersen graph is class two
w
v
u
There are at least two red edges on the inner cycle.
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The Petersen graph is class two
The outer 5−cycle is colored with three colors
v
There are at least two red edges, two blue edges, and two green edgeson the inner cycle.
The inner cycle must have at least 6 edges.
The inner cycle is a 5-cycle and only has five edges!
This proves that the Petersen graph is not 3-edge colorable and thusit is class two.
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The Petersen graph is class two
The outer 5−cycle is colored with three colors
v
There are at least two red edges, two blue edges, and two green edgeson the inner cycle.
The inner cycle must have at least 6 edges.
The inner cycle is a 5-cycle and only has five edges!
This proves that the Petersen graph is not 3-edge colorable and thusit is class two.
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The Petersen graph is class two
The outer 5−cycle is colored with three colors
v
There are at least two red edges, two blue edges, and two green edgeson the inner cycle.
The inner cycle must have at least 6 edges.
The inner cycle is a 5-cycle and only has five edges!
This proves that the Petersen graph is not 3-edge colorable and thusit is class two.
Rong Luo (WVU) Edge Coloring of Graphs 75 / 105
The Petersen graph is class two
The outer 5−cycle is colored with three colors
v
There are at least two red edges, two blue edges, and two green edgeson the inner cycle.
The inner cycle must have at least 6 edges.
The inner cycle is a 5-cycle and only has five edges!
This proves that the Petersen graph is not 3-edge colorable and thusit is class two.Rong Luo (WVU) Edge Coloring of Graphs 75 / 105
Critical Graphs
G is critical (or ∆-critical) if χe(G ) = ∆ + 1 and χe(G − e) ≤ ∆ forany edge e in G .
2-critical graphs are odd cycles.
Criticality is a general concept in graph theory and can be definedwith respect to various graph parameters.
The importance of the notion of criticality is that problems for graphsin general may often be reduced to problems for critical graphs whosestructure is more restricted.
Critical graphs (with respect to the vertex chromatic number) werefirst introduced and used by Dirac in 1951.
Rong Luo (WVU) Edge Coloring of Graphs 76 / 105
Critical Graphs
G is critical (or ∆-critical) if χe(G ) = ∆ + 1 and χe(G − e) ≤ ∆ forany edge e in G .
2-critical graphs are odd cycles.
Criticality is a general concept in graph theory and can be definedwith respect to various graph parameters.
The importance of the notion of criticality is that problems for graphsin general may often be reduced to problems for critical graphs whosestructure is more restricted.
Critical graphs (with respect to the vertex chromatic number) werefirst introduced and used by Dirac in 1951.
Rong Luo (WVU) Edge Coloring of Graphs 76 / 105
Critical Graphs
G is critical (or ∆-critical) if χe(G ) = ∆ + 1 and χe(G − e) ≤ ∆ forany edge e in G .
2-critical graphs are odd cycles.
Criticality is a general concept in graph theory and can be definedwith respect to various graph parameters.
The importance of the notion of criticality is that problems for graphsin general may often be reduced to problems for critical graphs whosestructure is more restricted.
Critical graphs (with respect to the vertex chromatic number) werefirst introduced and used by Dirac in 1951.
Rong Luo (WVU) Edge Coloring of Graphs 76 / 105
Critical Graphs
G is critical (or ∆-critical) if χe(G ) = ∆ + 1 and χe(G − e) ≤ ∆ forany edge e in G .
2-critical graphs are odd cycles.
Criticality is a general concept in graph theory and can be definedwith respect to various graph parameters.
The importance of the notion of criticality is that problems for graphsin general may often be reduced to problems for critical graphs whosestructure is more restricted.
Critical graphs (with respect to the vertex chromatic number) werefirst introduced and used by Dirac in 1951.
Rong Luo (WVU) Edge Coloring of Graphs 76 / 105
Critical Graphs
G is critical (or ∆-critical) if χe(G ) = ∆ + 1 and χe(G − e) ≤ ∆ forany edge e in G .
2-critical graphs are odd cycles.
Criticality is a general concept in graph theory and can be definedwith respect to various graph parameters.
The importance of the notion of criticality is that problems for graphsin general may often be reduced to problems for critical graphs whosestructure is more restricted.
Critical graphs (with respect to the vertex chromatic number) werefirst introduced and used by Dirac in 1951.
Rong Luo (WVU) Edge Coloring of Graphs 76 / 105
Independence Number
An independent set is a set of vertices in which no pair of vertices areconnected with an edge. The size of a maximum independent set is calledindependence number.
v
u
x
y
The red vertices u, v , x , y form an independent set.The independence number of the Petersen graph is 4.
Rong Luo (WVU) Edge Coloring of Graphs 77 / 105
Example
A 2-factor of a graph is a 2-regular spanning subgraph.
v
u
x
y
The blue cycles form a 2-factor.
Rong Luo (WVU) Edge Coloring of Graphs 78 / 105
2-factor
A 2-factor of a graph is a 2-regular spanning subgraph.
The blue cycles form a 2-factor.
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Hamiltonian cycle
A hamiltonian cycle of a graph is a cycle that contains all the vertices ofthe graph.
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Hamiltonian cycle
A hamiltonian cycle of a graph is a cycle that contains all the vertices ofthe graph.
Rong Luo (WVU) Edge Coloring of Graphs 81 / 105
Hamiltonian cycle
A hamiltonian cycle of a graph is a cycle that contains all thevertices of the graph.
A hamiltonian cycle is a special 2-factor.
A graph is hamiltonian if it has a hamiltonian cycle.
Rong Luo (WVU) Edge Coloring of Graphs 82 / 105
Hamiltonian cycle
A hamiltonian cycle of a graph is a cycle that contains all thevertices of the graph.
A hamiltonian cycle is a special 2-factor.
A graph is hamiltonian if it has a hamiltonian cycle.
Rong Luo (WVU) Edge Coloring of Graphs 82 / 105
Hamiltonian cycle
A hamiltonian cycle of a graph is a cycle that contains all thevertices of the graph.
A hamiltonian cycle is a special 2-factor.
A graph is hamiltonian if it has a hamiltonian cycle.
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Vizing’s Four Conjectures
In later 1960s, Vizing proposed the following four conjectures.
(Vizing’s Independence Number Conjecture) The independencenumber of a critical graph is at most half of the number of vertices.
(Vizing’s 2-factor Conjecture) Every critical graph has a 2-factor.
(Vizing’s Conjecture on the Size of Critical Graphs) The averagedegree of a critical graph is at least ∆− 1 + 3
n .
(Vizing’s Planar Graph Conjecture) Every planar graph withmaximum degree 6 or 7 is class one.
Rong Luo (WVU) Edge Coloring of Graphs 83 / 105
Vizing’s Four Conjectures
In later 1960s, Vizing proposed the following four conjectures.
(Vizing’s Independence Number Conjecture) The independencenumber of a critical graph is at most half of the number of vertices.
(Vizing’s 2-factor Conjecture) Every critical graph has a 2-factor.
(Vizing’s Conjecture on the Size of Critical Graphs) The averagedegree of a critical graph is at least ∆− 1 + 3
n .
(Vizing’s Planar Graph Conjecture) Every planar graph withmaximum degree 6 or 7 is class one.
Rong Luo (WVU) Edge Coloring of Graphs 83 / 105
Vizing’s Four Conjectures
In later 1960s, Vizing proposed the following four conjectures.
(Vizing’s Independence Number Conjecture) The independencenumber of a critical graph is at most half of the number of vertices.
(Vizing’s 2-factor Conjecture) Every critical graph has a 2-factor.
(Vizing’s Conjecture on the Size of Critical Graphs) The averagedegree of a critical graph is at least ∆− 1 + 3
n .
(Vizing’s Planar Graph Conjecture) Every planar graph withmaximum degree 6 or 7 is class one.
Rong Luo (WVU) Edge Coloring of Graphs 83 / 105
Vizing’s Four Conjectures
In later 1960s, Vizing proposed the following four conjectures.
(Vizing’s Independence Number Conjecture) The independencenumber of a critical graph is at most half of the number of vertices.
(Vizing’s 2-factor Conjecture) Every critical graph has a 2-factor.
(Vizing’s Conjecture on the Size of Critical Graphs) The averagedegree of a critical graph is at least ∆− 1 + 3
n .
(Vizing’s Planar Graph Conjecture) Every planar graph withmaximum degree 6 or 7 is class one.
Rong Luo (WVU) Edge Coloring of Graphs 83 / 105
History
Chapter 9, Twenty Pretty Edge Coloring Conjectures
Sample
9 (late 1960s), 6 (1970s), 4 (early 1980s), 1 (1990)
Rong Luo (WVU) Edge Coloring of Graphs 85 / 105
History
Chapter 9, Twenty Pretty Edge Coloring Conjectures
Sample
9 (late 1960s), 6 (1970s), 4 (early 1980s), 1 (1990)
Rong Luo (WVU) Edge Coloring of Graphs 85 / 105
History
Tashkinov, 2000, introduced a method, called Tashkinov tree, whichis a generalization of Vizing Fan and Kiearstead path.
Three Ph.D dissertations on edge coloring (Goldberg Conjecture)
Kurt O. (2009) The Ohio State University, (Neil Robertson)
MacDonald, J. (2009), University of Waterloo (Penny Haxell)
Scheide (2007), ( Michael Stiebitz)
Rong Luo (WVU) Edge Coloring of Graphs 86 / 105
History
Tashkinov, 2000, introduced a method, called Tashkinov tree, whichis a generalization of Vizing Fan and Kiearstead path.
Three Ph.D dissertations on edge coloring (Goldberg Conjecture)
Kurt O. (2009) The Ohio State University, (Neil Robertson)
MacDonald, J. (2009), University of Waterloo (Penny Haxell)
Scheide (2007), ( Michael Stiebitz)
Rong Luo (WVU) Edge Coloring of Graphs 86 / 105
History
Tashkinov, 2000, introduced a method, called Tashkinov tree, whichis a generalization of Vizing Fan and Kiearstead path.
Three Ph.D dissertations on edge coloring (Goldberg Conjecture)
Kurt O. (2009) The Ohio State University, (Neil Robertson)
MacDonald, J. (2009), University of Waterloo (Penny Haxell)
Scheide (2007), ( Michael Stiebitz)
Rong Luo (WVU) Edge Coloring of Graphs 86 / 105
History
Tashkinov, 2000, introduced a method, called Tashkinov tree, whichis a generalization of Vizing Fan and Kiearstead path.
Three Ph.D dissertations on edge coloring (Goldberg Conjecture)
Kurt O. (2009) The Ohio State University, (Neil Robertson)
MacDonald, J. (2009), University of Waterloo (Penny Haxell)
Scheide (2007), ( Michael Stiebitz)
Rong Luo (WVU) Edge Coloring of Graphs 86 / 105
History
Tashkinov, 2000, introduced a method, called Tashkinov tree, whichis a generalization of Vizing Fan and Kiearstead path.
Three Ph.D dissertations on edge coloring (Goldberg Conjecture)
Kurt O. (2009) The Ohio State University, (Neil Robertson)
MacDonald, J. (2009), University of Waterloo (Penny Haxell)
Scheide (2007), ( Michael Stiebitz)
Rong Luo (WVU) Edge Coloring of Graphs 86 / 105
Edge chromatic number
χe(G ) ≥ ∆.
How high can edge chromatic number be?
Theorem
(Konig, 1916) If G is a bipartite graph, then χe(G ) = ∆.
Konig’s result is the first result on edge coloring with a correct proof.
Theorem
(Vizing Theorem)For each simple graph G , χe(G ) = ∆ or ∆ + 1.
For a simple graph G , χe(G ) ∈ {∆,∆ + 1}.
Rong Luo (WVU) Edge Coloring of Graphs 87 / 105
Edge chromatic number
χe(G ) ≥ ∆.
How high can edge chromatic number be?
Theorem
(Konig, 1916) If G is a bipartite graph, then χe(G ) = ∆.
Konig’s result is the first result on edge coloring with a correct proof.
Theorem
(Vizing Theorem)For each simple graph G , χe(G ) = ∆ or ∆ + 1.
For a simple graph G , χe(G ) ∈ {∆,∆ + 1}.
Rong Luo (WVU) Edge Coloring of Graphs 87 / 105
Edge chromatic number
χe(G ) ≥ ∆.
How high can edge chromatic number be?
Theorem
(Konig, 1916) If G is a bipartite graph, then χe(G ) = ∆.
Konig’s result is the first result on edge coloring with a correct proof.
Theorem
(Vizing Theorem)For each simple graph G , χe(G ) = ∆ or ∆ + 1.
For a simple graph G , χe(G ) ∈ {∆,∆ + 1}.
Rong Luo (WVU) Edge Coloring of Graphs 87 / 105
Edge chromatic number
χe(G ) ≥ ∆.
How high can edge chromatic number be?
Theorem
(Konig, 1916) If G is a bipartite graph, then χe(G ) = ∆.
Konig’s result is the first result on edge coloring with a correct proof.
Theorem
(Vizing Theorem)For each simple graph G , χe(G ) = ∆ or ∆ + 1.
For a simple graph G , χe(G ) ∈ {∆,∆ + 1}.
Rong Luo (WVU) Edge Coloring of Graphs 87 / 105
Edge chromatic number
χe(G ) ≥ ∆.
How high can edge chromatic number be?
Theorem
(Konig, 1916) If G is a bipartite graph, then χe(G ) = ∆.
Konig’s result is the first result on edge coloring with a correct proof.
Theorem
(Vizing Theorem)For each simple graph G , χe(G ) = ∆ or ∆ + 1.
For a simple graph G , χe(G ) ∈ {∆,∆ + 1}.
Rong Luo (WVU) Edge Coloring of Graphs 87 / 105
Edge chromatic number
χe(G ) ≥ ∆.
How high can edge chromatic number be?
Theorem
(Konig, 1916) If G is a bipartite graph, then χe(G ) = ∆.
Konig’s result is the first result on edge coloring with a correct proof.
Theorem
(Vizing Theorem)For each simple graph G , χe(G ) = ∆ or ∆ + 1.
For a simple graph G , χe(G ) ∈ {∆,∆ + 1}.
Rong Luo (WVU) Edge Coloring of Graphs 87 / 105
Edge chromatic number–Upper bounds
χe(G ) ≤ 3∆2 (Shannon, 1949).
χe(G ) ≤ ∆ + µ where µ is the multiplicity of G (Vizing, 1964).
χe(G ) ≤ ∆ + d∆−2go−1e (Goldberg, 1970s).
χe(G ) ≤ ∆ + d µb g
2ce (Stephen, 2001).
In general χe(G ) ∈ {∆,∆ + 1, · · · ,∆ + µ}.For a simple graph G , χe(G ) ∈ {∆,∆ + 1}.
Rong Luo (WVU) Edge Coloring of Graphs 89 / 105
Edge chromatic number–Upper bounds
χe(G ) ≤ 3∆2 (Shannon, 1949).
χe(G ) ≤ ∆ + µ where µ is the multiplicity of G (Vizing, 1964).
χe(G ) ≤ ∆ + d∆−2go−1e (Goldberg, 1970s).
χe(G ) ≤ ∆ + d µb g
2ce (Stephen, 2001).
In general χe(G ) ∈ {∆,∆ + 1, · · · ,∆ + µ}.For a simple graph G , χe(G ) ∈ {∆,∆ + 1}.
Rong Luo (WVU) Edge Coloring of Graphs 89 / 105
Edge chromatic number–Upper bounds
χe(G ) ≤ 3∆2 (Shannon, 1949).
χe(G ) ≤ ∆ + µ where µ is the multiplicity of G (Vizing, 1964).
χe(G ) ≤ ∆ + d∆−2go−1e (Goldberg, 1970s).
χe(G ) ≤ ∆ + d µb g
2ce (Stephen, 2001).
In general χe(G ) ∈ {∆,∆ + 1, · · · ,∆ + µ}.For a simple graph G , χe(G ) ∈ {∆,∆ + 1}.
Rong Luo (WVU) Edge Coloring of Graphs 89 / 105
Edge chromatic number–Upper bounds
χe(G ) ≤ 3∆2 (Shannon, 1949).
χe(G ) ≤ ∆ + µ where µ is the multiplicity of G (Vizing, 1964).
χe(G ) ≤ ∆ + d∆−2go−1e (Goldberg, 1970s).
χe(G ) ≤ ∆ + d µb g
2ce (Stephen, 2001).
In general χe(G ) ∈ {∆,∆ + 1, · · · ,∆ + µ}.For a simple graph G , χe(G ) ∈ {∆,∆ + 1}.
Rong Luo (WVU) Edge Coloring of Graphs 89 / 105
Edge chromatic number–Upper bounds
χe(G ) ≤ 3∆2 (Shannon, 1949).
χe(G ) ≤ ∆ + µ where µ is the multiplicity of G (Vizing, 1964).
χe(G ) ≤ ∆ + d∆−2go−1e (Goldberg, 1970s).
χe(G ) ≤ ∆ + d µb g
2ce (Stephen, 2001).
In general χe(G ) ∈ {∆,∆ + 1, · · · ,∆ + µ}.
For a simple graph G , χe(G ) ∈ {∆,∆ + 1}.
Rong Luo (WVU) Edge Coloring of Graphs 89 / 105
Edge chromatic number–Upper bounds
χe(G ) ≤ 3∆2 (Shannon, 1949).
χe(G ) ≤ ∆ + µ where µ is the multiplicity of G (Vizing, 1964).
χe(G ) ≤ ∆ + d∆−2go−1e (Goldberg, 1970s).
χe(G ) ≤ ∆ + d µb g
2ce (Stephen, 2001).
In general χe(G ) ∈ {∆,∆ + 1, · · · ,∆ + µ}.For a simple graph G , χe(G ) ∈ {∆,∆ + 1}.
Rong Luo (WVU) Edge Coloring of Graphs 89 / 105
Edge chromatic number–Another nontrivial lower bound
Suppose G has an edge k-coloring with the color classes E1,E2, · · ·Ek
where k = χe(G )
Each color class is a matching, so |Ei | ≤ b |V (G)|2 c.
The total number of edges in G , |E (G )| ≤ kb |V (G)|2 c.
χe(G ) = k ≥ d |E(G)|b |V (G)|
2ce.
For any subgraph H of G , we have χe(G ) ≥ χe(H) ≥ d |E(H)|b |V (H)|
2ce.
χe(G ) ≥ maxH⊆G ,|V (H)|≥2
d |E (H)|b |V (H)|
2 ce
Rong Luo (WVU) Edge Coloring of Graphs 90 / 105
Edge chromatic number–Another nontrivial lower bound
Suppose G has an edge k-coloring with the color classes E1,E2, · · ·Ek
where k = χe(G )
Each color class is a matching, so |Ei | ≤ b |V (G)|2 c.
The total number of edges in G , |E (G )| ≤ kb |V (G)|2 c.
χe(G ) = k ≥ d |E(G)|b |V (G)|
2ce.
For any subgraph H of G , we have χe(G ) ≥ χe(H) ≥ d |E(H)|b |V (H)|
2ce.
χe(G ) ≥ maxH⊆G ,|V (H)|≥2
d |E (H)|b |V (H)|
2 ce
Rong Luo (WVU) Edge Coloring of Graphs 90 / 105
Edge chromatic number–Another nontrivial lower bound
Suppose G has an edge k-coloring with the color classes E1,E2, · · ·Ek
where k = χe(G )
Each color class is a matching, so |Ei | ≤ b |V (G)|2 c.
The total number of edges in G , |E (G )| ≤ kb |V (G)|2 c.
χe(G ) = k ≥ d |E(G)|b |V (G)|
2ce.
For any subgraph H of G , we have χe(G ) ≥ χe(H) ≥ d |E(H)|b |V (H)|
2ce.
χe(G ) ≥ maxH⊆G ,|V (H)|≥2
d |E (H)|b |V (H)|
2 ce
Rong Luo (WVU) Edge Coloring of Graphs 90 / 105
Edge chromatic number–Another nontrivial lower bound
Suppose G has an edge k-coloring with the color classes E1,E2, · · ·Ek
where k = χe(G )
Each color class is a matching, so |Ei | ≤ b |V (G)|2 c.
The total number of edges in G , |E (G )| ≤ kb |V (G)|2 c.
χe(G ) = k ≥ d |E(G)|b |V (G)|
2ce.
For any subgraph H of G , we have χe(G ) ≥ χe(H) ≥ d |E(H)|b |V (H)|
2ce.
χe(G ) ≥ maxH⊆G ,|V (H)|≥2
d |E (H)|b |V (H)|
2 ce
Rong Luo (WVU) Edge Coloring of Graphs 90 / 105
Edge chromatic number–Another nontrivial lower bound
Suppose G has an edge k-coloring with the color classes E1,E2, · · ·Ek
where k = χe(G )
Each color class is a matching, so |Ei | ≤ b |V (G)|2 c.
The total number of edges in G , |E (G )| ≤ kb |V (G)|2 c.
χe(G ) = k ≥ d |E(G)|b |V (G)|
2ce.
For any subgraph H of G , we have χe(G ) ≥ χe(H) ≥ d |E(H)|b |V (H)|
2ce.
χe(G ) ≥ maxH⊆G ,|V (H)|≥2
d |E (H)|b |V (H)|
2 ce
Rong Luo (WVU) Edge Coloring of Graphs 90 / 105
Edge chromatic number–Another nontrivial lower bound
Suppose G has an edge k-coloring with the color classes E1,E2, · · ·Ek
where k = χe(G )
Each color class is a matching, so |Ei | ≤ b |V (G)|2 c.
The total number of edges in G , |E (G )| ≤ kb |V (G)|2 c.
χe(G ) = k ≥ d |E(G)|b |V (G)|
2ce.
For any subgraph H of G , we have χe(G ) ≥ χe(H) ≥ d |E(H)|b |V (H)|
2ce.
χe(G ) ≥ maxH⊆G ,|V (H)|≥2
d |E (H)|b |V (H)|
2 ce
Rong Luo (WVU) Edge Coloring of Graphs 90 / 105
Edge chromatic number–Another nontrivial lower bound
χe(G ) ≥ maxH⊆G ,|V (H)|≥2
d |E (H)|b |V (H)
2 |ce
w(G ) = maxH⊆G ,|V (H)|≥2
d |E (H)|b |V (H)|
2 ce
w(G ) is called the density of G .
χe(G ) ≥ w(G )
χe(G ) ≥ max{∆,w(G )}.The maximum value can be achieved when |V (H)| is odd.
w(G ) = maxH⊆G ,|V (H)|≥2
d |E (H)|b |V (H)|
2 ce = max
H⊆G ,|V (H)|≥3,oddd 2|E (H)||V (H)| − 1
e
Rong Luo (WVU) Edge Coloring of Graphs 91 / 105
Edge chromatic number–Another nontrivial lower bound
χe(G ) ≥ maxH⊆G ,|V (H)|≥2
d |E (H)|b |V (H)
2 |ce
w(G ) = maxH⊆G ,|V (H)|≥2
d |E (H)|b |V (H)|
2 ce
w(G ) is called the density of G .
χe(G ) ≥ w(G )
χe(G ) ≥ max{∆,w(G )}.The maximum value can be achieved when |V (H)| is odd.
w(G ) = maxH⊆G ,|V (H)|≥2
d |E (H)|b |V (H)|
2 ce = max
H⊆G ,|V (H)|≥3,oddd 2|E (H)||V (H)| − 1
e
Rong Luo (WVU) Edge Coloring of Graphs 91 / 105
Edge chromatic number–Another nontrivial lower bound
χe(G ) ≥ maxH⊆G ,|V (H)|≥2
d |E (H)|b |V (H)
2 |ce
w(G ) = maxH⊆G ,|V (H)|≥2
d |E (H)|b |V (H)|
2 ce
w(G ) is called the density of G .
χe(G ) ≥ w(G )
χe(G ) ≥ max{∆,w(G )}.The maximum value can be achieved when |V (H)| is odd.
w(G ) = maxH⊆G ,|V (H)|≥2
d |E (H)|b |V (H)|
2 ce = max
H⊆G ,|V (H)|≥3,oddd 2|E (H)||V (H)| − 1
e
Rong Luo (WVU) Edge Coloring of Graphs 91 / 105
Edge chromatic number–Another nontrivial lower bound
χe(G ) ≥ maxH⊆G ,|V (H)|≥2
d |E (H)|b |V (H)
2 |ce
w(G ) = maxH⊆G ,|V (H)|≥2
d |E (H)|b |V (H)|
2 ce
w(G ) is called the density of G .
χe(G ) ≥ w(G )
χe(G ) ≥ max{∆,w(G )}.The maximum value can be achieved when |V (H)| is odd.
w(G ) = maxH⊆G ,|V (H)|≥2
d |E (H)|b |V (H)|
2 ce = max
H⊆G ,|V (H)|≥3,oddd 2|E (H)||V (H)| − 1
e
Rong Luo (WVU) Edge Coloring of Graphs 91 / 105
Edge chromatic number–Another nontrivial lower bound
χe(G ) ≥ maxH⊆G ,|V (H)|≥2
d |E (H)|b |V (H)
2 |ce
w(G ) = maxH⊆G ,|V (H)|≥2
d |E (H)|b |V (H)|
2 ce
w(G ) is called the density of G .
χe(G ) ≥ w(G )
χe(G ) ≥ max{∆,w(G )}.
The maximum value can be achieved when |V (H)| is odd.
w(G ) = maxH⊆G ,|V (H)|≥2
d |E (H)|b |V (H)|
2 ce = max
H⊆G ,|V (H)|≥3,oddd 2|E (H)||V (H)| − 1
e
Rong Luo (WVU) Edge Coloring of Graphs 91 / 105
Edge chromatic number–Another nontrivial lower bound
χe(G ) ≥ maxH⊆G ,|V (H)|≥2
d |E (H)|b |V (H)
2 |ce
w(G ) = maxH⊆G ,|V (H)|≥2
d |E (H)|b |V (H)|
2 ce
w(G ) is called the density of G .
χe(G ) ≥ w(G )
χe(G ) ≥ max{∆,w(G )}.The maximum value can be achieved when |V (H)| is odd.
w(G ) = maxH⊆G ,|V (H)|≥2
d |E (H)|b |V (H)|
2 ce = max
H⊆G ,|V (H)|≥3,oddd 2|E (H)||V (H)| − 1
e
Rong Luo (WVU) Edge Coloring of Graphs 91 / 105
Edge chromatic number–Another nontrivial lower bound
χe(G ) ≥ maxH⊆G ,|V (H)|≥2
d |E (H)|b |V (H)
2 |ce
w(G ) = maxH⊆G ,|V (H)|≥2
d |E (H)|b |V (H)|
2 ce
w(G ) is called the density of G .
χe(G ) ≥ w(G )
χe(G ) ≥ max{∆,w(G )}.The maximum value can be achieved when |V (H)| is odd.
w(G ) = maxH⊆G ,|V (H)|≥2
d |E (H)|b |V (H)|
2 ce = max
H⊆G ,|V (H)|≥3,oddd 2|E (H)||V (H)| − 1
e
Rong Luo (WVU) Edge Coloring of Graphs 91 / 105
Seymour’s r -graph Conjecture
For simple graphs, ∆ ≤ χe(G ) ≤ ∆ + 1.
In general, χe(G ) ≥ max{∆,w(G )}.Is it true χe(G ) ≤ max{∆,w(G )}+ 1.?
Conjecture
(Seymour’s r -graph Conjecture, 1979) Let G be a graph. Then
χe(G ) ≤ max{∆,w(G )}+ 1.
Rong Luo (WVU) Edge Coloring of Graphs 92 / 105
Seymour’s r -graph Conjecture
For simple graphs, ∆ ≤ χe(G ) ≤ ∆ + 1.
In general, χe(G ) ≥ max{∆,w(G )}.
Is it true χe(G ) ≤ max{∆,w(G )}+ 1.?
Conjecture
(Seymour’s r -graph Conjecture, 1979) Let G be a graph. Then
χe(G ) ≤ max{∆,w(G )}+ 1.
Rong Luo (WVU) Edge Coloring of Graphs 92 / 105
Seymour’s r -graph Conjecture
For simple graphs, ∆ ≤ χe(G ) ≤ ∆ + 1.
In general, χe(G ) ≥ max{∆,w(G )}.Is it true χe(G ) ≤ max{∆,w(G )}+ 1.?
Conjecture
(Seymour’s r -graph Conjecture, 1979) Let G be a graph. Then
χe(G ) ≤ max{∆,w(G )}+ 1.
Rong Luo (WVU) Edge Coloring of Graphs 92 / 105
Seymour’s r -graph Conjecture
For simple graphs, ∆ ≤ χe(G ) ≤ ∆ + 1.
In general, χe(G ) ≥ max{∆,w(G )}.Is it true χe(G ) ≤ max{∆,w(G )}+ 1.?
Conjecture
(Seymour’s r -graph Conjecture, 1979) Let G be a graph. Then
χe(G ) ≤ max{∆,w(G )}+ 1.
Rong Luo (WVU) Edge Coloring of Graphs 92 / 105
Goldberg Conjecture
Conjecture
(Goldberg Conjecture) Let G be a graph. Then
χe(G ) ≤ max{∆ + 1,w(G )}.
Goldberg conjecture was proposed by Goldberg in 1970 andindependently by Seymour in 1979.
Goldberg Conjecture is equivalent to the following statements.
Let G be a graph. If χe(G ) ≥ ∆ + 2, then χe(G ) = w(G ).
χe(G ) ∈ {∆,∆ + 1,w(G )}.
Rong Luo (WVU) Edge Coloring of Graphs 93 / 105
Goldberg Conjecture
Conjecture
(Goldberg Conjecture) Let G be a graph. Then
χe(G ) ≤ max{∆ + 1,w(G )}.
Goldberg conjecture was proposed by Goldberg in 1970 andindependently by Seymour in 1979.
Goldberg Conjecture is equivalent to the following statements.
Let G be a graph. If χe(G ) ≥ ∆ + 2, then χe(G ) = w(G ).
χe(G ) ∈ {∆,∆ + 1,w(G )}.
Rong Luo (WVU) Edge Coloring of Graphs 93 / 105
Goldberg Conjecture
Conjecture
(Goldberg Conjecture) Let G be a graph. Then
χe(G ) ≤ max{∆ + 1,w(G )}.
Goldberg conjecture was proposed by Goldberg in 1970 andindependently by Seymour in 1979.
Goldberg Conjecture is equivalent to the following statements.
Let G be a graph. If χe(G ) ≥ ∆ + 2, then χe(G ) = w(G ).
χe(G ) ∈ {∆,∆ + 1,w(G )}.
Rong Luo (WVU) Edge Coloring of Graphs 93 / 105
Goldberg Conjecture
Conjecture
(Goldberg Conjecture) Let G be a graph. Then
χe(G ) ≤ max{∆ + 1,w(G )}.
Goldberg conjecture was proposed by Goldberg in 1970 andindependently by Seymour in 1979.
Goldberg Conjecture is equivalent to the following statements.
Let G be a graph. If χe(G ) ≥ ∆ + 2, then χe(G ) = w(G ).
χe(G ) ∈ {∆,∆ + 1,w(G )}.
Rong Luo (WVU) Edge Coloring of Graphs 93 / 105
Goldberg Conjecture
Conjecture
(Goldberg Conjecture) Let G be a graph. Then
χe(G ) ≤ max{∆ + 1,w(G )}.
Goldberg conjecture was proposed by Goldberg in 1970 andindependently by Seymour in 1979.
Goldberg Conjecture is equivalent to the following statements.
Let G be a graph. If χe(G ) ≥ ∆ + 2, then χe(G ) = w(G ).
χe(G ) ∈ {∆,∆ + 1,w(G )}.
Rong Luo (WVU) Edge Coloring of Graphs 93 / 105
Goldberg Conjecture–Importance
Goldberg Conjecture implies that if χe(G ) ≥ ∆ + 2, then there is apolynomial algorithm to compute χe(G ).
So it implies that the difficulty in determining χe(G ) is only todistinguish between two cases χe(G ) = ∆ and χe(G ) = ∆ + 1, whichis NP-hard proved by Holyer in 1980.
Goldberg Conjecture also implies Seymour’s r -graph conjecture andJakobsen’s critical graph conjecture.
G is critical if χe(G − e) < χe(G ) for any edge e.
Rong Luo (WVU) Edge Coloring of Graphs 94 / 105
Goldberg Conjecture–Importance
Goldberg Conjecture implies that if χe(G ) ≥ ∆ + 2, then there is apolynomial algorithm to compute χe(G ).
So it implies that the difficulty in determining χe(G ) is only todistinguish between two cases χe(G ) = ∆ and χe(G ) = ∆ + 1, whichis NP-hard proved by Holyer in 1980.
Goldberg Conjecture also implies Seymour’s r -graph conjecture andJakobsen’s critical graph conjecture.
G is critical if χe(G − e) < χe(G ) for any edge e.
Rong Luo (WVU) Edge Coloring of Graphs 94 / 105
Goldberg Conjecture–Importance
Goldberg Conjecture implies that if χe(G ) ≥ ∆ + 2, then there is apolynomial algorithm to compute χe(G ).
So it implies that the difficulty in determining χe(G ) is only todistinguish between two cases χe(G ) = ∆ and χe(G ) = ∆ + 1, whichis NP-hard proved by Holyer in 1980.
Goldberg Conjecture also implies Seymour’s r -graph conjecture andJakobsen’s critical graph conjecture.
G is critical if χe(G − e) < χe(G ) for any edge e.
Rong Luo (WVU) Edge Coloring of Graphs 94 / 105
Goldberg Conjecture–Importance
Goldberg Conjecture implies that if χe(G ) ≥ ∆ + 2, then there is apolynomial algorithm to compute χe(G ).
So it implies that the difficulty in determining χe(G ) is only todistinguish between two cases χe(G ) = ∆ and χe(G ) = ∆ + 1, whichis NP-hard proved by Holyer in 1980.
Goldberg Conjecture also implies Seymour’s r -graph conjecture andJakobsen’s critical graph conjecture.
G is critical if χe(G − e) < χe(G ) for any edge e.
Rong Luo (WVU) Edge Coloring of Graphs 94 / 105
Seymour’s r -graph Conjecture–Original Version
A r -regular graph is an r -graph if for every set X ⊆ V (G ) with |X |odd, |∂G (X )| ≥ r .
Conjecture
(Seymour’s r -graph Conjecture, 1979) Every r -graph satisfiesχe(G ) ≤ r + 1.
w(G ) ≤ r for each r -graph.Let H ⊆ G with |V (H)| odd. Let X = V (H). Then2|E (H)| ≤ r |X | − r = r(|X | − 1)
2|E(H)||V (H)|−1 ≤
r(|X |−1)|X |−1 = r .
If an r -regular graph has an edge r -coloring, then it must be anr -graph. (Why?)Seymour proved that every graph with max{∆,w(G )} ≤ r iscontained in an r -graph as a subgraph.Since w(G ) ≤ r for each r -graph, Goldberg Conjecture impliesχe(G ) ≤ max{∆ + 1,w(G )} = r + 1.
Rong Luo (WVU) Edge Coloring of Graphs 95 / 105
Seymour’s r -graph Conjecture–Original Version
A r -regular graph is an r -graph if for every set X ⊆ V (G ) with |X |odd, |∂G (X )| ≥ r .
Conjecture
(Seymour’s r -graph Conjecture, 1979) Every r -graph satisfiesχe(G ) ≤ r + 1.
w(G ) ≤ r for each r -graph.Let H ⊆ G with |V (H)| odd. Let X = V (H). Then2|E (H)| ≤ r |X | − r = r(|X | − 1)
2|E(H)||V (H)|−1 ≤
r(|X |−1)|X |−1 = r .
If an r -regular graph has an edge r -coloring, then it must be anr -graph. (Why?)Seymour proved that every graph with max{∆,w(G )} ≤ r iscontained in an r -graph as a subgraph.Since w(G ) ≤ r for each r -graph, Goldberg Conjecture impliesχe(G ) ≤ max{∆ + 1,w(G )} = r + 1.
Rong Luo (WVU) Edge Coloring of Graphs 95 / 105
Seymour’s r -graph Conjecture–Original Version
A r -regular graph is an r -graph if for every set X ⊆ V (G ) with |X |odd, |∂G (X )| ≥ r .
Conjecture
(Seymour’s r -graph Conjecture, 1979) Every r -graph satisfiesχe(G ) ≤ r + 1.
w(G ) ≤ r for each r -graph.
Let H ⊆ G with |V (H)| odd. Let X = V (H). Then2|E (H)| ≤ r |X | − r = r(|X | − 1)
2|E(H)||V (H)|−1 ≤
r(|X |−1)|X |−1 = r .
If an r -regular graph has an edge r -coloring, then it must be anr -graph. (Why?)Seymour proved that every graph with max{∆,w(G )} ≤ r iscontained in an r -graph as a subgraph.Since w(G ) ≤ r for each r -graph, Goldberg Conjecture impliesχe(G ) ≤ max{∆ + 1,w(G )} = r + 1.
Rong Luo (WVU) Edge Coloring of Graphs 95 / 105
Seymour’s r -graph Conjecture–Original Version
A r -regular graph is an r -graph if for every set X ⊆ V (G ) with |X |odd, |∂G (X )| ≥ r .
Conjecture
(Seymour’s r -graph Conjecture, 1979) Every r -graph satisfiesχe(G ) ≤ r + 1.
w(G ) ≤ r for each r -graph.Let H ⊆ G with |V (H)| odd. Let X = V (H). Then2|E (H)| ≤ r |X | − r = r(|X | − 1)
2|E(H)||V (H)|−1 ≤
r(|X |−1)|X |−1 = r .
If an r -regular graph has an edge r -coloring, then it must be anr -graph. (Why?)Seymour proved that every graph with max{∆,w(G )} ≤ r iscontained in an r -graph as a subgraph.Since w(G ) ≤ r for each r -graph, Goldberg Conjecture impliesχe(G ) ≤ max{∆ + 1,w(G )} = r + 1.
Rong Luo (WVU) Edge Coloring of Graphs 95 / 105
Seymour’s r -graph Conjecture–Original Version
A r -regular graph is an r -graph if for every set X ⊆ V (G ) with |X |odd, |∂G (X )| ≥ r .
Conjecture
(Seymour’s r -graph Conjecture, 1979) Every r -graph satisfiesχe(G ) ≤ r + 1.
w(G ) ≤ r for each r -graph.Let H ⊆ G with |V (H)| odd. Let X = V (H). Then2|E (H)| ≤ r |X | − r = r(|X | − 1)
2|E(H)||V (H)|−1 ≤
r(|X |−1)|X |−1 = r .
If an r -regular graph has an edge r -coloring, then it must be anr -graph. (Why?)Seymour proved that every graph with max{∆,w(G )} ≤ r iscontained in an r -graph as a subgraph.Since w(G ) ≤ r for each r -graph, Goldberg Conjecture impliesχe(G ) ≤ max{∆ + 1,w(G )} = r + 1.
Rong Luo (WVU) Edge Coloring of Graphs 95 / 105
Seymour’s r -graph Conjecture–Original Version
A r -regular graph is an r -graph if for every set X ⊆ V (G ) with |X |odd, |∂G (X )| ≥ r .
Conjecture
(Seymour’s r -graph Conjecture, 1979) Every r -graph satisfiesχe(G ) ≤ r + 1.
w(G ) ≤ r for each r -graph.Let H ⊆ G with |V (H)| odd. Let X = V (H). Then2|E (H)| ≤ r |X | − r = r(|X | − 1)
2|E(H)||V (H)|−1 ≤
r(|X |−1)|X |−1 = r .
If an r -regular graph has an edge r -coloring, then it must be anr -graph. (Why?)
Seymour proved that every graph with max{∆,w(G )} ≤ r iscontained in an r -graph as a subgraph.Since w(G ) ≤ r for each r -graph, Goldberg Conjecture impliesχe(G ) ≤ max{∆ + 1,w(G )} = r + 1.
Rong Luo (WVU) Edge Coloring of Graphs 95 / 105
Seymour’s r -graph Conjecture–Original Version
A r -regular graph is an r -graph if for every set X ⊆ V (G ) with |X |odd, |∂G (X )| ≥ r .
Conjecture
(Seymour’s r -graph Conjecture, 1979) Every r -graph satisfiesχe(G ) ≤ r + 1.
w(G ) ≤ r for each r -graph.Let H ⊆ G with |V (H)| odd. Let X = V (H). Then2|E (H)| ≤ r |X | − r = r(|X | − 1)
2|E(H)||V (H)|−1 ≤
r(|X |−1)|X |−1 = r .
If an r -regular graph has an edge r -coloring, then it must be anr -graph. (Why?)Seymour proved that every graph with max{∆,w(G )} ≤ r iscontained in an r -graph as a subgraph.
Since w(G ) ≤ r for each r -graph, Goldberg Conjecture impliesχe(G ) ≤ max{∆ + 1,w(G )} = r + 1.
Rong Luo (WVU) Edge Coloring of Graphs 95 / 105
Seymour’s r -graph Conjecture–Original Version
A r -regular graph is an r -graph if for every set X ⊆ V (G ) with |X |odd, |∂G (X )| ≥ r .
Conjecture
(Seymour’s r -graph Conjecture, 1979) Every r -graph satisfiesχe(G ) ≤ r + 1.
w(G ) ≤ r for each r -graph.Let H ⊆ G with |V (H)| odd. Let X = V (H). Then2|E (H)| ≤ r |X | − r = r(|X | − 1)
2|E(H)||V (H)|−1 ≤
r(|X |−1)|X |−1 = r .
If an r -regular graph has an edge r -coloring, then it must be anr -graph. (Why?)Seymour proved that every graph with max{∆,w(G )} ≤ r iscontained in an r -graph as a subgraph.Since w(G ) ≤ r for each r -graph, Goldberg Conjecture impliesχe(G ) ≤ max{∆ + 1,w(G )} = r + 1.Rong Luo (WVU) Edge Coloring of Graphs 95 / 105
Seymour’s r -graph Conjecture–Equivalent Version
Seymour’s r -graph conjecture is equivalent to
Every graph G satisfies χe(G ) ≤ max{∆,w(G )}+ 1.
Seymour’s r -graph conjecture is true for r ≤ 15.
Seymour’s r -graph conjecture suggests that if G is an r -graph then,for all t ≥ 1, either χe(tG ) = max{∆(tG ),w(tG )} orχe(tG ) = max{∆(tG ),w(tG )}+ 1.
Rong Luo (WVU) Edge Coloring of Graphs 96 / 105
Seymour’s r -graph Conjecture–Equivalent Version
Seymour’s r -graph conjecture is equivalent to
Every graph G satisfies χe(G ) ≤ max{∆,w(G )}+ 1.
Seymour’s r -graph conjecture is true for r ≤ 15.
Seymour’s r -graph conjecture suggests that if G is an r -graph then,for all t ≥ 1, either χe(tG ) = max{∆(tG ),w(tG )} orχe(tG ) = max{∆(tG ),w(tG )}+ 1.
Rong Luo (WVU) Edge Coloring of Graphs 96 / 105
Seymour’s r -graph Conjecture–Equivalent Version
Seymour’s r -graph conjecture is equivalent to
Every graph G satisfies χe(G ) ≤ max{∆,w(G )}+ 1.
Seymour’s r -graph conjecture is true for r ≤ 15.
Seymour’s r -graph conjecture suggests that if G is an r -graph then,for all t ≥ 1, either χe(tG ) = max{∆(tG ),w(tG )} orχe(tG ) = max{∆(tG ),w(tG )}+ 1.
Rong Luo (WVU) Edge Coloring of Graphs 96 / 105
Seymour’s r -graph Conjecture–Equivalent Version
Seymour’s r -graph conjecture is equivalent to
Every graph G satisfies χe(G ) ≤ max{∆,w(G )}+ 1.
Seymour’s r -graph conjecture is true for r ≤ 15.
Seymour’s r -graph conjecture suggests that if G is an r -graph then,for all t ≥ 1, either χe(tG ) = max{∆(tG ),w(tG )} orχe(tG ) = max{∆(tG ),w(tG )}+ 1.
Rong Luo (WVU) Edge Coloring of Graphs 96 / 105
Seymour’s Exact Conjecture
For planar graph, Seymour proposed a stronger conjecture.
Conjecture
(Seymour’s Exact Conjecture, 1979) Every planar graph G satisfiesχe(G ) = max{∆,w(G )}.
The cases r = 0, 1, 2 are trivial.
Seymour proved the exact conjecture for series-parrallel graphs in1990.
Marcotte verified the conjecture for the class of graphs not containingK3,3 or K−5 as a minor, 2001.
The case r = 3 is equivalent to the Four Color Theorem.
The cases r = 4, 5 was proved by Guenin (2011).
Dvorak, Kawarabayashi, and Kral solved the case r = 6 in 2011
The case of k = 7 was proved by Edwards and Kawarabayashi in 2012.
Chudnovsky, Edwards and Seymour solved the case k = 8 in 2012.
Rong Luo (WVU) Edge Coloring of Graphs 97 / 105
Seymour’s Exact Conjecture
For planar graph, Seymour proposed a stronger conjecture.
Conjecture
(Seymour’s Exact Conjecture, 1979) Every planar graph G satisfiesχe(G ) = max{∆,w(G )}.
The cases r = 0, 1, 2 are trivial.
Seymour proved the exact conjecture for series-parrallel graphs in1990.
Marcotte verified the conjecture for the class of graphs not containingK3,3 or K−5 as a minor, 2001.
The case r = 3 is equivalent to the Four Color Theorem.
The cases r = 4, 5 was proved by Guenin (2011).
Dvorak, Kawarabayashi, and Kral solved the case r = 6 in 2011
The case of k = 7 was proved by Edwards and Kawarabayashi in 2012.
Chudnovsky, Edwards and Seymour solved the case k = 8 in 2012.
Rong Luo (WVU) Edge Coloring of Graphs 97 / 105
Seymour’s Exact Conjecture
For planar graph, Seymour proposed a stronger conjecture.
Conjecture
(Seymour’s Exact Conjecture, 1979) Every planar graph G satisfiesχe(G ) = max{∆,w(G )}.
The cases r = 0, 1, 2 are trivial.
Seymour proved the exact conjecture for series-parrallel graphs in1990.
Marcotte verified the conjecture for the class of graphs not containingK3,3 or K−5 as a minor, 2001.
The case r = 3 is equivalent to the Four Color Theorem.
The cases r = 4, 5 was proved by Guenin (2011).
Dvorak, Kawarabayashi, and Kral solved the case r = 6 in 2011
The case of k = 7 was proved by Edwards and Kawarabayashi in 2012.
Chudnovsky, Edwards and Seymour solved the case k = 8 in 2012.
Rong Luo (WVU) Edge Coloring of Graphs 97 / 105
Seymour’s Exact Conjecture
For planar graph, Seymour proposed a stronger conjecture.
Conjecture
(Seymour’s Exact Conjecture, 1979) Every planar graph G satisfiesχe(G ) = max{∆,w(G )}.
The cases r = 0, 1, 2 are trivial.
Seymour proved the exact conjecture for series-parrallel graphs in1990.
Marcotte verified the conjecture for the class of graphs not containingK3,3 or K−5 as a minor, 2001.
The case r = 3 is equivalent to the Four Color Theorem.
The cases r = 4, 5 was proved by Guenin (2011).
Dvorak, Kawarabayashi, and Kral solved the case r = 6 in 2011
The case of k = 7 was proved by Edwards and Kawarabayashi in 2012.
Chudnovsky, Edwards and Seymour solved the case k = 8 in 2012.
Rong Luo (WVU) Edge Coloring of Graphs 97 / 105
Seymour’s Exact Conjecture
For planar graph, Seymour proposed a stronger conjecture.
Conjecture
(Seymour’s Exact Conjecture, 1979) Every planar graph G satisfiesχe(G ) = max{∆,w(G )}.
The cases r = 0, 1, 2 are trivial.
Seymour proved the exact conjecture for series-parrallel graphs in1990.
Marcotte verified the conjecture for the class of graphs not containingK3,3 or K−5 as a minor, 2001.
The case r = 3 is equivalent to the Four Color Theorem.
The cases r = 4, 5 was proved by Guenin (2011).
Dvorak, Kawarabayashi, and Kral solved the case r = 6 in 2011
The case of k = 7 was proved by Edwards and Kawarabayashi in 2012.
Chudnovsky, Edwards and Seymour solved the case k = 8 in 2012.
Rong Luo (WVU) Edge Coloring of Graphs 97 / 105
Seymour’s Exact Conjecture
For planar graph, Seymour proposed a stronger conjecture.
Conjecture
(Seymour’s Exact Conjecture, 1979) Every planar graph G satisfiesχe(G ) = max{∆,w(G )}.
The cases r = 0, 1, 2 are trivial.
Seymour proved the exact conjecture for series-parrallel graphs in1990.
Marcotte verified the conjecture for the class of graphs not containingK3,3 or K−5 as a minor, 2001.
The case r = 3 is equivalent to the Four Color Theorem.
The cases r = 4, 5 was proved by Guenin (2011).
Dvorak, Kawarabayashi, and Kral solved the case r = 6 in 2011
The case of k = 7 was proved by Edwards and Kawarabayashi in 2012.
Chudnovsky, Edwards and Seymour solved the case k = 8 in 2012.
Rong Luo (WVU) Edge Coloring of Graphs 97 / 105
Seymour’s Exact Conjecture
For planar graph, Seymour proposed a stronger conjecture.
Conjecture
(Seymour’s Exact Conjecture, 1979) Every planar graph G satisfiesχe(G ) = max{∆,w(G )}.
The cases r = 0, 1, 2 are trivial.
Seymour proved the exact conjecture for series-parrallel graphs in1990.
Marcotte verified the conjecture for the class of graphs not containingK3,3 or K−5 as a minor, 2001.
The case r = 3 is equivalent to the Four Color Theorem.
The cases r = 4, 5 was proved by Guenin (2011).
Dvorak, Kawarabayashi, and Kral solved the case r = 6 in 2011
The case of k = 7 was proved by Edwards and Kawarabayashi in 2012.
Chudnovsky, Edwards and Seymour solved the case k = 8 in 2012.
Rong Luo (WVU) Edge Coloring of Graphs 97 / 105
Seymour’s Exact Conjecture
For planar graph, Seymour proposed a stronger conjecture.
Conjecture
(Seymour’s Exact Conjecture, 1979) Every planar graph G satisfiesχe(G ) = max{∆,w(G )}.
The cases r = 0, 1, 2 are trivial.
Seymour proved the exact conjecture for series-parrallel graphs in1990.
Marcotte verified the conjecture for the class of graphs not containingK3,3 or K−5 as a minor, 2001.
The case r = 3 is equivalent to the Four Color Theorem.
The cases r = 4, 5 was proved by Guenin (2011).
Dvorak, Kawarabayashi, and Kral solved the case r = 6 in 2011
The case of k = 7 was proved by Edwards and Kawarabayashi in 2012.
Chudnovsky, Edwards and Seymour solved the case k = 8 in 2012.
Rong Luo (WVU) Edge Coloring of Graphs 97 / 105
Seymour’s Exact Conjecture
For planar graph, Seymour proposed a stronger conjecture.
Conjecture
(Seymour’s Exact Conjecture, 1979) Every planar graph G satisfiesχe(G ) = max{∆,w(G )}.
The cases r = 0, 1, 2 are trivial.
Seymour proved the exact conjecture for series-parrallel graphs in1990.
Marcotte verified the conjecture for the class of graphs not containingK3,3 or K−5 as a minor, 2001.
The case r = 3 is equivalent to the Four Color Theorem.
The cases r = 4, 5 was proved by Guenin (2011).
Dvorak, Kawarabayashi, and Kral solved the case r = 6 in 2011
The case of k = 7 was proved by Edwards and Kawarabayashi in 2012.
Chudnovsky, Edwards and Seymour solved the case k = 8 in 2012.
Rong Luo (WVU) Edge Coloring of Graphs 97 / 105
Seymour’s Exact Conjecture
For planar graph, Seymour proposed a stronger conjecture.
Conjecture
(Seymour’s Exact Conjecture, 1979) Every planar graph G satisfiesχe(G ) = max{∆,w(G )}.
The cases r = 0, 1, 2 are trivial.
Seymour proved the exact conjecture for series-parrallel graphs in1990.
Marcotte verified the conjecture for the class of graphs not containingK3,3 or K−5 as a minor, 2001.
The case r = 3 is equivalent to the Four Color Theorem.
The cases r = 4, 5 was proved by Guenin (2011).
Dvorak, Kawarabayashi, and Kral solved the case r = 6 in 2011
The case of k = 7 was proved by Edwards and Kawarabayashi in 2012.
Chudnovsky, Edwards and Seymour solved the case k = 8 in 2012.
Rong Luo (WVU) Edge Coloring of Graphs 97 / 105
Jakobsen’s critical graph conjecture
Conjecture
(Jakobosen’s Critical Graph Conjecture, 1973) Let G be a critical graph,and let
χe(G ) >m
m − 1∆(G ) +
m − 3
m − 1.
for an odd integer m ≥ 3. Then |V (G )| ≤ m − 2.
Jakobsen’s conjecture is trivial for m = 3 by Shannons upper bound.
Anderson proved that Goldberg’s Conjecture implies Jakobsen’sconjecture.
Jakobsen’s conjecture was verified for 5 ≤ m ≤ 15.
Rong Luo (WVU) Edge Coloring of Graphs 98 / 105
Jakobsen’s critical graph conjecture
Conjecture
(Jakobosen’s Critical Graph Conjecture, 1973) Let G be a critical graph,and let
χe(G ) >m
m − 1∆(G ) +
m − 3
m − 1.
for an odd integer m ≥ 3. Then |V (G )| ≤ m − 2.
Jakobsen’s conjecture is trivial for m = 3 by Shannons upper bound.
Anderson proved that Goldberg’s Conjecture implies Jakobsen’sconjecture.
Jakobsen’s conjecture was verified for 5 ≤ m ≤ 15.
Rong Luo (WVU) Edge Coloring of Graphs 98 / 105
Jakobsen’s critical graph conjecture
Conjecture
(Jakobosen’s Critical Graph Conjecture, 1973) Let G be a critical graph,and let
χe(G ) >m
m − 1∆(G ) +
m − 3
m − 1.
for an odd integer m ≥ 3. Then |V (G )| ≤ m − 2.
Jakobsen’s conjecture is trivial for m = 3 by Shannons upper bound.
Anderson proved that Goldberg’s Conjecture implies Jakobsen’sconjecture.
Jakobsen’s conjecture was verified for 5 ≤ m ≤ 15.
Rong Luo (WVU) Edge Coloring of Graphs 98 / 105
Jakobsen’s critical graph conjecture
Conjecture
(Jakobosen’s Critical Graph Conjecture, 1973) Let G be a critical graph,and let
χe(G ) >m
m − 1∆(G ) +
m − 3
m − 1.
for an odd integer m ≥ 3. Then |V (G )| ≤ m − 2.
Jakobsen’s conjecture is trivial for m = 3 by Shannons upper bound.
Anderson proved that Goldberg’s Conjecture implies Jakobsen’sconjecture.
Jakobsen’s conjecture was verified for 5 ≤ m ≤ 15.
Rong Luo (WVU) Edge Coloring of Graphs 98 / 105
Fractional edge chromatic number
An edge coloring can be considered as an Integer ProgrammingProblem.
Let M denote the set of all matchings of a graph G .
For each edge e, let Me denote the set of all matchings containing e.
χe(G ) = min∑
M∈MyM ,
subject to:(1)
∑M∈Me
yM = 1 for each edge e ∈ E (G ).(2) yM ∈ {0, 1}
Rong Luo (WVU) Edge Coloring of Graphs 99 / 105
Fractional edge chromatic number
An edge coloring can be considered as an Integer ProgrammingProblem.
Let M denote the set of all matchings of a graph G .
For each edge e, let Me denote the set of all matchings containing e.
χe(G ) = min∑
M∈MyM ,
subject to:(1)
∑M∈Me
yM = 1 for each edge e ∈ E (G ).(2) yM ∈ {0, 1}
Rong Luo (WVU) Edge Coloring of Graphs 99 / 105
Fractional edge chromatic number
An edge coloring can be considered as an Integer ProgrammingProblem.
Let M denote the set of all matchings of a graph G .
For each edge e, let Me denote the set of all matchings containing e.
χe(G ) = min∑
M∈MyM ,
subject to:(1)
∑M∈Me
yM = 1 for each edge e ∈ E (G ).(2) yM ∈ {0, 1}
Rong Luo (WVU) Edge Coloring of Graphs 99 / 105
Fractional edge chromatic number
An edge coloring can be considered as an Integer ProgrammingProblem.
Let M denote the set of all matchings of a graph G .
For each edge e, let Me denote the set of all matchings containing e.
χe(G ) = min∑
M∈MyM ,
subject to:(1)
∑M∈Me
yM = 1 for each edge e ∈ E (G ).(2) yM ∈ {0, 1}
Rong Luo (WVU) Edge Coloring of Graphs 99 / 105
Fractional chromatic index
The fractional edge chromatic number χ∗e(G ) is defined as:
χ∗e(G ) = min∑
M∈MyM ,
subject to:(1)
∑M∈Me
yM = 1 for each edge e ∈ E (G ).(2) 0 ≤ yM ≤ 1
With matching techniques one can compute the fractional edgechromatic number in polynomial time.
From Edmond’s matching polytope theorem,
χ∗e(G ) = max{∆, maxH⊆G ,|V (H)|≥2
|E (H)|b1
2 |V (H)|c}.
If χ∗e(G ) = ∆, then w(G ) ≤ ∆
If χ∗e(G ) > ∆, then w(G ) = dχ∗e(G )e and w(G ) can be computed inpolynomial time.
Rong Luo (WVU) Edge Coloring of Graphs 100 / 105
Fractional chromatic index
The fractional edge chromatic number χ∗e(G ) is defined as:
χ∗e(G ) = min∑
M∈MyM ,
subject to:(1)
∑M∈Me
yM = 1 for each edge e ∈ E (G ).(2) 0 ≤ yM ≤ 1
With matching techniques one can compute the fractional edgechromatic number in polynomial time.
From Edmond’s matching polytope theorem,
χ∗e(G ) = max{∆, maxH⊆G ,|V (H)|≥2
|E (H)|b1
2 |V (H)|c}.
If χ∗e(G ) = ∆, then w(G ) ≤ ∆
If χ∗e(G ) > ∆, then w(G ) = dχ∗e(G )e and w(G ) can be computed inpolynomial time.
Rong Luo (WVU) Edge Coloring of Graphs 100 / 105
Fractional chromatic index
The fractional edge chromatic number χ∗e(G ) is defined as:
χ∗e(G ) = min∑
M∈MyM ,
subject to:(1)
∑M∈Me
yM = 1 for each edge e ∈ E (G ).(2) 0 ≤ yM ≤ 1
With matching techniques one can compute the fractional edgechromatic number in polynomial time.
From Edmond’s matching polytope theorem,
χ∗e(G ) = max{∆, maxH⊆G ,|V (H)|≥2
|E (H)|b1
2 |V (H)|c}.
If χ∗e(G ) = ∆, then w(G ) ≤ ∆
If χ∗e(G ) > ∆, then w(G ) = dχ∗e(G )e and w(G ) can be computed inpolynomial time.
Rong Luo (WVU) Edge Coloring of Graphs 100 / 105
Fractional chromatic index
The fractional edge chromatic number χ∗e(G ) is defined as:
χ∗e(G ) = min∑
M∈MyM ,
subject to:(1)
∑M∈Me
yM = 1 for each edge e ∈ E (G ).(2) 0 ≤ yM ≤ 1
With matching techniques one can compute the fractional edgechromatic number in polynomial time.
From Edmond’s matching polytope theorem,
χ∗e(G ) = max{∆, maxH⊆G ,|V (H)|≥2
|E (H)|b1
2 |V (H)|c}.
If χ∗e(G ) = ∆, then w(G ) ≤ ∆
If χ∗e(G ) > ∆, then w(G ) = dχ∗e(G )e and w(G ) can be computed inpolynomial time.
Rong Luo (WVU) Edge Coloring of Graphs 100 / 105
Fractional chromatic index
The fractional edge chromatic number χ∗e(G ) is defined as:
χ∗e(G ) = min∑
M∈MyM ,
subject to:(1)
∑M∈Me
yM = 1 for each edge e ∈ E (G ).(2) 0 ≤ yM ≤ 1
With matching techniques one can compute the fractional edgechromatic number in polynomial time.
From Edmond’s matching polytope theorem,
χ∗e(G ) = max{∆, maxH⊆G ,|V (H)|≥2
|E (H)|b1
2 |V (H)|c}.
If χ∗e(G ) = ∆, then w(G ) ≤ ∆
If χ∗e(G ) > ∆, then w(G ) = dχ∗e(G )e and w(G ) can be computed inpolynomial time.
Rong Luo (WVU) Edge Coloring of Graphs 100 / 105
Fractional chromatic index
The fractional edge chromatic number χ∗e(G ) is defined as:
χ∗e(G ) = min∑
M∈MyM ,
subject to:(1)
∑M∈Me
yM = 1 for each edge e ∈ E (G ).(2) 0 ≤ yM ≤ 1
With matching techniques one can compute the fractional edgechromatic number in polynomial time.
From Edmond’s matching polytope theorem,
χ∗e(G ) = max{∆, maxH⊆G ,|V (H)|≥2
|E (H)|b1
2 |V (H)|c}.
If χ∗e(G ) = ∆, then w(G ) ≤ ∆
If χ∗e(G ) > ∆, then w(G ) = dχ∗e(G )e and w(G ) can be computed inpolynomial time.
Rong Luo (WVU) Edge Coloring of Graphs 100 / 105
Fractional chromatic index
The computation of the edge chromatic number χe(G ) is NP-hard
The fractional edge chromatic number can be computed inpolynomial time.
It is not clear whether the density w(G ) can be computed inpolynomial time.
max{∆(G ),w(G )} can be computed in polynomial time
Goldberg Conjecture is equivalent to the claim that χe(G ) = dχ∗e(G )efor every graph G with χe(G ) ≥ ∆ + 2.
χe(G ) ≤ χ∗e(G ) +√χ∗e(G )/2 (Schiede, Sanders and Steuer 9/2 ).
Kahn proved in 1996 that every graph G satisfies χe(G ) ∼ χ∗e(G ) asχ∗e(G )→∞.
Rong Luo (WVU) Edge Coloring of Graphs 101 / 105
Fractional chromatic index
The computation of the edge chromatic number χe(G ) is NP-hard
The fractional edge chromatic number can be computed inpolynomial time.
It is not clear whether the density w(G ) can be computed inpolynomial time.
max{∆(G ),w(G )} can be computed in polynomial time
Goldberg Conjecture is equivalent to the claim that χe(G ) = dχ∗e(G )efor every graph G with χe(G ) ≥ ∆ + 2.
χe(G ) ≤ χ∗e(G ) +√χ∗e(G )/2 (Schiede, Sanders and Steuer 9/2 ).
Kahn proved in 1996 that every graph G satisfies χe(G ) ∼ χ∗e(G ) asχ∗e(G )→∞.
Rong Luo (WVU) Edge Coloring of Graphs 101 / 105
Fractional chromatic index
The computation of the edge chromatic number χe(G ) is NP-hard
The fractional edge chromatic number can be computed inpolynomial time.
It is not clear whether the density w(G ) can be computed inpolynomial time.
max{∆(G ),w(G )} can be computed in polynomial time
Goldberg Conjecture is equivalent to the claim that χe(G ) = dχ∗e(G )efor every graph G with χe(G ) ≥ ∆ + 2.
χe(G ) ≤ χ∗e(G ) +√χ∗e(G )/2 (Schiede, Sanders and Steuer 9/2 ).
Kahn proved in 1996 that every graph G satisfies χe(G ) ∼ χ∗e(G ) asχ∗e(G )→∞.
Rong Luo (WVU) Edge Coloring of Graphs 101 / 105
Fractional chromatic index
The computation of the edge chromatic number χe(G ) is NP-hard
The fractional edge chromatic number can be computed inpolynomial time.
It is not clear whether the density w(G ) can be computed inpolynomial time.
max{∆(G ),w(G )} can be computed in polynomial time
Goldberg Conjecture is equivalent to the claim that χe(G ) = dχ∗e(G )efor every graph G with χe(G ) ≥ ∆ + 2.
χe(G ) ≤ χ∗e(G ) +√χ∗e(G )/2 (Schiede, Sanders and Steuer 9/2 ).
Kahn proved in 1996 that every graph G satisfies χe(G ) ∼ χ∗e(G ) asχ∗e(G )→∞.
Rong Luo (WVU) Edge Coloring of Graphs 101 / 105
Fractional chromatic index
The computation of the edge chromatic number χe(G ) is NP-hard
The fractional edge chromatic number can be computed inpolynomial time.
It is not clear whether the density w(G ) can be computed inpolynomial time.
max{∆(G ),w(G )} can be computed in polynomial time
Goldberg Conjecture is equivalent to the claim that χe(G ) = dχ∗e(G )efor every graph G with χe(G ) ≥ ∆ + 2.
χe(G ) ≤ χ∗e(G ) +√χ∗e(G )/2 (Schiede, Sanders and Steuer 9/2 ).
Kahn proved in 1996 that every graph G satisfies χe(G ) ∼ χ∗e(G ) asχ∗e(G )→∞.
Rong Luo (WVU) Edge Coloring of Graphs 101 / 105
Fractional chromatic index
The computation of the edge chromatic number χe(G ) is NP-hard
The fractional edge chromatic number can be computed inpolynomial time.
It is not clear whether the density w(G ) can be computed inpolynomial time.
max{∆(G ),w(G )} can be computed in polynomial time
Goldberg Conjecture is equivalent to the claim that χe(G ) = dχ∗e(G )efor every graph G with χe(G ) ≥ ∆ + 2.
χe(G ) ≤ χ∗e(G ) +√χ∗e(G )/2 (Schiede, Sanders and Steuer 9/2 ).
Kahn proved in 1996 that every graph G satisfies χe(G ) ∼ χ∗e(G ) asχ∗e(G )→∞.
Rong Luo (WVU) Edge Coloring of Graphs 101 / 105
Fractional chromatic index
The computation of the edge chromatic number χe(G ) is NP-hard
The fractional edge chromatic number can be computed inpolynomial time.
It is not clear whether the density w(G ) can be computed inpolynomial time.
max{∆(G ),w(G )} can be computed in polynomial time
Goldberg Conjecture is equivalent to the claim that χe(G ) = dχ∗e(G )efor every graph G with χe(G ) ≥ ∆ + 2.
χe(G ) ≤ χ∗e(G ) +√χ∗e(G )/2 (Schiede, Sanders and Steuer 9/2 ).
Kahn proved in 1996 that every graph G satisfies χe(G ) ∼ χ∗e(G ) asχ∗e(G )→∞.
Rong Luo (WVU) Edge Coloring of Graphs 101 / 105
Progress toward Goldberg’s Conjecture
Goldberg’s conjecture has been verified for the following cases:
(Goldberg, 1974; Anderson, 1977) χe(G ) > 54 ∆ + 2
4 .
( Anderson, 1977) χe(G ) > 76 ∆ + 4
6 .
(Goldberg, 1984) χe(G ) > 98 ∆ + 6
8 .
(Nishizeki, Kashiwagi, 1990; Tashkinov, 2000) χe(G ) > 1110 ∆ + 8
10 .
(Favrholdt, Stiebitz, Toft, 2006) χe(G ) > 1312 ∆ + 10
12 .
(Scheide, Stiebitz, 2009) χe(G ) > 1514 ∆ + 12
14 .
(Chen, Yu, Zang, 2011; Schied, 2010) χe(G ) > ∆(G ) +√
∆2 .
(Kurt, 2009) χe(G ) > ∆(G ) + 3
√∆2 .
Rong Luo (WVU) Edge Coloring of Graphs 102 / 105
Progress toward Goldberg’s Conjecture
Goldberg’s conjecture has been verified for the following cases:
(Goldberg, 1974; Anderson, 1977) χe(G ) > 54 ∆ + 2
4 .
( Anderson, 1977) χe(G ) > 76 ∆ + 4
6 .
(Goldberg, 1984) χe(G ) > 98 ∆ + 6
8 .
(Nishizeki, Kashiwagi, 1990; Tashkinov, 2000) χe(G ) > 1110 ∆ + 8
10 .
(Favrholdt, Stiebitz, Toft, 2006) χe(G ) > 1312 ∆ + 10
12 .
(Scheide, Stiebitz, 2009) χe(G ) > 1514 ∆ + 12
14 .
(Chen, Yu, Zang, 2011; Schied, 2010) χe(G ) > ∆(G ) +√
∆2 .
(Kurt, 2009) χe(G ) > ∆(G ) + 3
√∆2 .
Rong Luo (WVU) Edge Coloring of Graphs 102 / 105
Progress toward Goldberg’s Conjecture
Goldberg’s conjecture has been verified for the following cases:
(Goldberg, 1974; Anderson, 1977) χe(G ) > 54 ∆ + 2
4 .
( Anderson, 1977) χe(G ) > 76 ∆ + 4
6 .
(Goldberg, 1984) χe(G ) > 98 ∆ + 6
8 .
(Nishizeki, Kashiwagi, 1990; Tashkinov, 2000) χe(G ) > 1110 ∆ + 8
10 .
(Favrholdt, Stiebitz, Toft, 2006) χe(G ) > 1312 ∆ + 10
12 .
(Scheide, Stiebitz, 2009) χe(G ) > 1514 ∆ + 12
14 .
(Chen, Yu, Zang, 2011; Schied, 2010) χe(G ) > ∆(G ) +√
∆2 .
(Kurt, 2009) χe(G ) > ∆(G ) + 3
√∆2 .
Rong Luo (WVU) Edge Coloring of Graphs 102 / 105
Progress toward Goldberg’s Conjecture
Goldberg’s conjecture has been verified for the following cases:
(Goldberg, 1974; Anderson, 1977) χe(G ) > 54 ∆ + 2
4 .
( Anderson, 1977) χe(G ) > 76 ∆ + 4
6 .
(Goldberg, 1984) χe(G ) > 98 ∆ + 6
8 .
(Nishizeki, Kashiwagi, 1990; Tashkinov, 2000) χe(G ) > 1110 ∆ + 8
10 .
(Favrholdt, Stiebitz, Toft, 2006) χe(G ) > 1312 ∆ + 10
12 .
(Scheide, Stiebitz, 2009) χe(G ) > 1514 ∆ + 12
14 .
(Chen, Yu, Zang, 2011; Schied, 2010) χe(G ) > ∆(G ) +√
∆2 .
(Kurt, 2009) χe(G ) > ∆(G ) + 3
√∆2 .
Rong Luo (WVU) Edge Coloring of Graphs 102 / 105
Progress toward Goldberg’s Conjecture
Goldberg’s conjecture has been verified for the following cases:
(Goldberg, 1974; Anderson, 1977) χe(G ) > 54 ∆ + 2
4 .
( Anderson, 1977) χe(G ) > 76 ∆ + 4
6 .
(Goldberg, 1984) χe(G ) > 98 ∆ + 6
8 .
(Nishizeki, Kashiwagi, 1990; Tashkinov, 2000) χe(G ) > 1110 ∆ + 8
10 .
(Favrholdt, Stiebitz, Toft, 2006) χe(G ) > 1312 ∆ + 10
12 .
(Scheide, Stiebitz, 2009) χe(G ) > 1514 ∆ + 12
14 .
(Chen, Yu, Zang, 2011; Schied, 2010) χe(G ) > ∆(G ) +√
∆2 .
(Kurt, 2009) χe(G ) > ∆(G ) + 3
√∆2 .
Rong Luo (WVU) Edge Coloring of Graphs 102 / 105
Progress toward Goldberg’s Conjecture
Goldberg’s conjecture has been verified for the following cases:
(Goldberg, 1974; Anderson, 1977) χe(G ) > 54 ∆ + 2
4 .
( Anderson, 1977) χe(G ) > 76 ∆ + 4
6 .
(Goldberg, 1984) χe(G ) > 98 ∆ + 6
8 .
(Nishizeki, Kashiwagi, 1990; Tashkinov, 2000) χe(G ) > 1110 ∆ + 8
10 .
(Favrholdt, Stiebitz, Toft, 2006) χe(G ) > 1312 ∆ + 10
12 .
(Scheide, Stiebitz, 2009) χe(G ) > 1514 ∆ + 12
14 .
(Chen, Yu, Zang, 2011; Schied, 2010) χe(G ) > ∆(G ) +√
∆2 .
(Kurt, 2009) χe(G ) > ∆(G ) + 3
√∆2 .
Rong Luo (WVU) Edge Coloring of Graphs 102 / 105
Progress toward Goldberg’s Conjecture
Goldberg’s conjecture has been verified for the following cases:
(Goldberg, 1974; Anderson, 1977) χe(G ) > 54 ∆ + 2
4 .
( Anderson, 1977) χe(G ) > 76 ∆ + 4
6 .
(Goldberg, 1984) χe(G ) > 98 ∆ + 6
8 .
(Nishizeki, Kashiwagi, 1990; Tashkinov, 2000) χe(G ) > 1110 ∆ + 8
10 .
(Favrholdt, Stiebitz, Toft, 2006) χe(G ) > 1312 ∆ + 10
12 .
(Scheide, Stiebitz, 2009) χe(G ) > 1514 ∆ + 12
14 .
(Chen, Yu, Zang, 2011; Schied, 2010) χe(G ) > ∆(G ) +√
∆2 .
(Kurt, 2009) χe(G ) > ∆(G ) + 3
√∆2 .
Rong Luo (WVU) Edge Coloring of Graphs 102 / 105
Progress toward Goldberg’s Conjecture
Goldberg’s conjecture has been verified for the following cases:
(Goldberg, 1974; Anderson, 1977) χe(G ) > 54 ∆ + 2
4 .
( Anderson, 1977) χe(G ) > 76 ∆ + 4
6 .
(Goldberg, 1984) χe(G ) > 98 ∆ + 6
8 .
(Nishizeki, Kashiwagi, 1990; Tashkinov, 2000) χe(G ) > 1110 ∆ + 8
10 .
(Favrholdt, Stiebitz, Toft, 2006) χe(G ) > 1312 ∆ + 10
12 .
(Scheide, Stiebitz, 2009) χe(G ) > 1514 ∆ + 12
14 .
(Chen, Yu, Zang, 2011; Schied, 2010) χe(G ) > ∆(G ) +√
∆2 .
(Kurt, 2009) χe(G ) > ∆(G ) + 3
√∆2 .
Rong Luo (WVU) Edge Coloring of Graphs 102 / 105
Vizing’s Four Conjectures
In later 1960s, Vizing proposed the following four conjectures for SIMPLEgraphs.
(Vizing’s Independence Number Conjecture) The independencenumber of a critical graph is at most half of the number of vertices.
(Vizing’s 2-factor Conjecture) Every critical graph has a 2-factor.
(Vizing’s Conjecture on the Size of Critical Graphs) The averagedegree of a critical graph is at least ∆− 1 + 3
n .
(Vizing’s Planar Graph Conjecture) Every planar graph withmaximum degree 6 or 7 is class one.
Rong Luo (WVU) Edge Coloring of Graphs 103 / 105
Vizing’s Four Conjectures
In later 1960s, Vizing proposed the following four conjectures for SIMPLEgraphs.
(Vizing’s Independence Number Conjecture) The independencenumber of a critical graph is at most half of the number of vertices.
(Vizing’s 2-factor Conjecture) Every critical graph has a 2-factor.
(Vizing’s Conjecture on the Size of Critical Graphs) The averagedegree of a critical graph is at least ∆− 1 + 3
n .
(Vizing’s Planar Graph Conjecture) Every planar graph withmaximum degree 6 or 7 is class one.
Rong Luo (WVU) Edge Coloring of Graphs 103 / 105
Vizing’s Four Conjectures
In later 1960s, Vizing proposed the following four conjectures for SIMPLEgraphs.
(Vizing’s Independence Number Conjecture) The independencenumber of a critical graph is at most half of the number of vertices.
(Vizing’s 2-factor Conjecture) Every critical graph has a 2-factor.
(Vizing’s Conjecture on the Size of Critical Graphs) The averagedegree of a critical graph is at least ∆− 1 + 3
n .
(Vizing’s Planar Graph Conjecture) Every planar graph withmaximum degree 6 or 7 is class one.
Rong Luo (WVU) Edge Coloring of Graphs 103 / 105
Vizing’s Four Conjectures
In later 1960s, Vizing proposed the following four conjectures for SIMPLEgraphs.
(Vizing’s Independence Number Conjecture) The independencenumber of a critical graph is at most half of the number of vertices.
(Vizing’s 2-factor Conjecture) Every critical graph has a 2-factor.
(Vizing’s Conjecture on the Size of Critical Graphs) The averagedegree of a critical graph is at least ∆− 1 + 3
n .
(Vizing’s Planar Graph Conjecture) Every planar graph withmaximum degree 6 or 7 is class one.
Rong Luo (WVU) Edge Coloring of Graphs 103 / 105
Other Conjectures
The Berge-Fulkerson Conjecture Every bridge less cubic graph Gcontains a family of six perfect matchings such that each edge of G iscontained in precisely two of the matchings.
(Equivalence) For each bridge less cubic graph G , χe(2G ) = 6.
(Berge’s conjecture) Every bridgeless cubic graph G contains a familyof five perfect matchings such that each edge of G is contained in atleast one of the matchings.
Mazzuoccolo proved that Berge’s conjecture is equivalent to TheBerge-Fulkerson Conjecture.
Matching cover:
mk(G ) = max{∪ki=1Mi ||E(G)| |M1,M2, · · · ,Mkare perfect matchings ofG}.
m1(G ) = 13 and Berge’s conjecture suggests that m5(G ) = 1.
Rong Luo (WVU) Edge Coloring of Graphs 104 / 105
Other Conjectures
The Berge-Fulkerson Conjecture Every bridge less cubic graph Gcontains a family of six perfect matchings such that each edge of G iscontained in precisely two of the matchings.
(Equivalence) For each bridge less cubic graph G , χe(2G ) = 6.
(Berge’s conjecture) Every bridgeless cubic graph G contains a familyof five perfect matchings such that each edge of G is contained in atleast one of the matchings.
Mazzuoccolo proved that Berge’s conjecture is equivalent to TheBerge-Fulkerson Conjecture.
Matching cover:
mk(G ) = max{∪ki=1Mi ||E(G)| |M1,M2, · · · ,Mkare perfect matchings ofG}.
m1(G ) = 13 and Berge’s conjecture suggests that m5(G ) = 1.
Rong Luo (WVU) Edge Coloring of Graphs 104 / 105
Other Conjectures
The Berge-Fulkerson Conjecture Every bridge less cubic graph Gcontains a family of six perfect matchings such that each edge of G iscontained in precisely two of the matchings.
(Equivalence) For each bridge less cubic graph G , χe(2G ) = 6.
(Berge’s conjecture) Every bridgeless cubic graph G contains a familyof five perfect matchings such that each edge of G is contained in atleast one of the matchings.
Mazzuoccolo proved that Berge’s conjecture is equivalent to TheBerge-Fulkerson Conjecture.
Matching cover:
mk(G ) = max{∪ki=1Mi ||E(G)| |M1,M2, · · · ,Mkare perfect matchings ofG}.
m1(G ) = 13 and Berge’s conjecture suggests that m5(G ) = 1.
Rong Luo (WVU) Edge Coloring of Graphs 104 / 105
Other Conjectures
The Berge-Fulkerson Conjecture Every bridge less cubic graph Gcontains a family of six perfect matchings such that each edge of G iscontained in precisely two of the matchings.
(Equivalence) For each bridge less cubic graph G , χe(2G ) = 6.
(Berge’s conjecture) Every bridgeless cubic graph G contains a familyof five perfect matchings such that each edge of G is contained in atleast one of the matchings.
Mazzuoccolo proved that Berge’s conjecture is equivalent to TheBerge-Fulkerson Conjecture.
Matching cover:
mk(G ) = max{∪ki=1Mi ||E(G)| |M1,M2, · · · ,Mkare perfect matchings ofG}.
m1(G ) = 13 and Berge’s conjecture suggests that m5(G ) = 1.
Rong Luo (WVU) Edge Coloring of Graphs 104 / 105
Other Conjectures
The Berge-Fulkerson Conjecture Every bridge less cubic graph Gcontains a family of six perfect matchings such that each edge of G iscontained in precisely two of the matchings.
(Equivalence) For each bridge less cubic graph G , χe(2G ) = 6.
(Berge’s conjecture) Every bridgeless cubic graph G contains a familyof five perfect matchings such that each edge of G is contained in atleast one of the matchings.
Mazzuoccolo proved that Berge’s conjecture is equivalent to TheBerge-Fulkerson Conjecture.
Matching cover:
mk(G ) = max{∪ki=1Mi ||E(G)| |M1,M2, · · · ,Mkare perfect matchings ofG}.
m1(G ) = 13 and Berge’s conjecture suggests that m5(G ) = 1.
Rong Luo (WVU) Edge Coloring of Graphs 104 / 105
Other Conjectures
The Berge-Fulkerson Conjecture Every bridge less cubic graph Gcontains a family of six perfect matchings such that each edge of G iscontained in precisely two of the matchings.
(Equivalence) For each bridge less cubic graph G , χe(2G ) = 6.
(Berge’s conjecture) Every bridgeless cubic graph G contains a familyof five perfect matchings such that each edge of G is contained in atleast one of the matchings.
Mazzuoccolo proved that Berge’s conjecture is equivalent to TheBerge-Fulkerson Conjecture.
Matching cover:
mk(G ) = max{∪ki=1Mi ||E(G)| |M1,M2, · · · ,Mkare perfect matchings ofG}.
m1(G ) = 13 and Berge’s conjecture suggests that m5(G ) = 1.
Rong Luo (WVU) Edge Coloring of Graphs 104 / 105
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