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

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

Edge Coloring of Graphs

Rong Luo

Department of MathematicsWest Virginia University

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.

Introduction

Example

Petersen graph

The degree of v is 3.The maximum degree ∆ = 3.

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.

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.

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.

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.

Graph Coloring

Example

Petersen graph

The degree of v is 3.The maximum degree ∆ = 3.

Example-Vertex Coloring-3-cloring of the Peterson Graph

Example–Edge Coloring

4−edge coloring of the petersen graph

Example-Edge Coloring

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.

Edge Coloring of Simple Graphs

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.

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.

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.

Coloring the map of USA

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.

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.

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.

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

Add A patchOrginal Map

Color the new map with 4 colors

Remove the patch Color the new map with 4 colors

(Cayley) Every plane map is face 4-colorable ⇐⇒ Every cubic plane mapis face 4-colorable.

Tait’s Theorem

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.

Tait’s Theorem

Theorem

Tait’s Theorem

Theorem

Face 4-coloring and Edge 3-coloring

Theorem

Edge 3−coloring

Face 4−coloring

From face 4-coloring to edge 3-coloring

{AB, CD} = RED

Face 4−coloring to Edge 3−coloring

{AB, CD} = RED

{AC,BD} = BLUE

{AB, CD} = RED

{AC,BD} = BLUE

{AD,BC} = GREEN

{AB, CD} = RED

{AC,BD} = BLUE

{AD,BC} = GREEN

{AB, CD} = RED

{AC,BD} = BLUE

• The edges are colored with THREE colors: Red, Blue, Green• Two edges sharing a common endvertex are colored with differentcolors(?)

{AD,BC} = GREEN

{AB, CD} = RED {AC,BD} = BLUE

{AD,BC} = GREEN

{AB, CD} = RED {AC,BD} = BLUE

• If a cubic plane graph has a face 4-coloring, then it has an edge3-coloring.

From edge 3-coloring to face 4-coloring

{AB, CD} = RED {AC, BD} = BLUE {AD, BC} = GREEN

Edge 3−coloring to Face 4−coloring

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 .

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 .

A natural lower bound

At least ∆ colors are needed.

At least ∆ colors are needed. So χe(G ) ≥ ∆. How big could χe(G ) be?

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.

Theorem

Examples

An odd cycle is class two.

Examples

An even cycle is class one.

Examples

An even cycle is class one.

The Petersen graph is class two

4−edge coloring of the petersen graph

Petersen graph

Suppose that we can color the edges of the Petersen graph with threecolors: red, blue, and green.

The outer cycle sees all three colors.

The outer 5−cycle is colored with three colors

Each vertex sees all 3 colors.

Each vertex sees all three colors

There are at least two red edges on the inner cycle.

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.

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.

Critical Graphs

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.

The red vertices u, v , x , y form an independent set.The independence number of the Petersen graph is 4.

Example

A 2-factor of a graph is a 2-regular spanning subgraph.

The blue cycles form a 2-factor.

2-factor

A 2-factor of a graph is a 2-regular spanning subgraph.

The blue cycles form a 2-factor.

Hamiltonian cycle

A hamiltonian cycle of a graph is a cycle that contains all the vertices ofthe graph.

Hamiltonian cycle

A hamiltonian cycle of a graph is a cycle that contains all the vertices ofthe graph.

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.

Hamiltonian cycle

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

(Vizing’s Planar Graph Conjecture) Every planar graph withmaximum degree 6 or 7 is class one.

Edge coloring of Multigraphs

History

Chapter 9, Twenty Pretty Edge Coloring Conjectures

Sample

9 (late 1960s), 6 (1970s), 4 (early 1980s), 1 (1990)

History

Chapter 9, Twenty Pretty Edge Coloring Conjectures

Sample

9 (late 1960s), 6 (1970s), 4 (early 1980s), 1 (1990)

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)

History

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}.

χe(G ) ≥ ∆.

Theorem

χe(G ) ≥ ∆.

Theorem

χe(G ) ≥ ∆.

Theorem

χe(G ) ≥ ∆.

Theorem

χe(G ) ≥ ∆.

Theorem

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}.

χe(G ) ≤ 3∆2 (Shannon, 1949).

χe(G ) ≤ ∆ + d µb g

χe(G ) ≤ 3∆2 (Shannon, 1949).

χe(G ) ≤ ∆ + d µb g

χe(G ) ≤ 3∆2 (Shannon, 1949).

χe(G ) ≤ ∆ + d µb g

χe(G ) ≤ 3∆2 (Shannon, 1949).

χe(G ) ≤ ∆ + d µb g

In general χe(G ) ∈ {∆,∆ + 1, · · · ,∆ + µ}.

χe(G ) ≤ 3∆2 (Shannon, 1949).

χe(G ) ≤ ∆ + d µb g

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)|

For any subgraph H of G , we have χe(G ) ≥ χe(H) ≥ d |E(H)|b |V (H)|

χe(G ) ≥ maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)|

where k = χe(G )

χe(G ) = k ≥ d |E(G)|b |V (G)|

χe(G ) ≥ maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)|

where k = χe(G )

χe(G ) = k ≥ d |E(G)|b |V (G)|

χe(G ) ≥ maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)|

where k = χe(G )

χe(G ) = k ≥ d |E(G)|b |V (G)|

χe(G ) ≥ maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)|

where k = χe(G )

χe(G ) = k ≥ d |E(G)|b |V (G)|

χe(G ) ≥ maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)|

where k = χe(G )

χe(G ) = k ≥ d |E(G)|b |V (G)|

χe(G ) ≥ maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)|

χe(G ) ≥ maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)

w(G ) = maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)|

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

χe(G ) ≥ maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)

w(G ) = maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)|

χe(G ) ≥ w(G )

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

χe(G ) ≥ maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)

w(G ) = maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)|

χe(G ) ≥ w(G )

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

χe(G ) ≥ maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)

w(G ) = maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)|

χe(G ) ≥ w(G )

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

χe(G ) ≥ maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)

w(G ) = maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)|

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

χe(G ) ≥ maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)

w(G ) = maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)|

χe(G ) ≥ w(G )

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

χe(G ) ≥ maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)

w(G ) = maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)|

χe(G ) ≥ w(G )

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

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.

In general, χe(G ) ≥ max{∆,w(G )}.

Is it true χe(G ) ≤ max{∆,w(G )}+ 1.?

Conjecture

χe(G ) ≤ max{∆,w(G )}+ 1.

Conjecture

χe(G ) ≤ max{∆,w(G )}+ 1.

Conjecture

χe(G ) ≤ max{∆,w(G )}+ 1.

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 )}.

Goldberg Conjecture

Conjecture

χe(G ) ≤ max{∆ + 1,w(G )}.

χe(G ) ∈ {∆,∆ + 1,w(G )}.

Goldberg Conjecture

Conjecture

χe(G ) ≤ max{∆ + 1,w(G )}.

χe(G ) ∈ {∆,∆ + 1,w(G )}.

Goldberg Conjecture

Conjecture

χe(G ) ≤ max{∆ + 1,w(G )}.

χe(G ) ∈ {∆,∆ + 1,w(G )}.

Goldberg Conjecture

Conjecture

χe(G ) ≤ max{∆ + 1,w(G )}.

χe(G ) ∈ {∆,∆ + 1,w(G )}.

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.

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.

Conjecture

2|E(H)||V (H)|−1 ≤

r(|X |−1)|X |−1 = r .

Conjecture

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 .

Conjecture

2|E(H)||V (H)|−1 ≤

r(|X |−1)|X |−1 = r .

Conjecture

2|E(H)||V (H)|−1 ≤

r(|X |−1)|X |−1 = r .

Conjecture

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.

Conjecture

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.

Conjecture

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.

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.

Conjecture

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.

Conjecture

χe(G ) >m

m − 1∆(G ) +

m − 3

m − 1.

Conjecture

χe(G ) >m

m − 1∆(G ) +

m − 3

m − 1.

Conjecture

χe(G ) >m

m − 1∆(G ) +

m − 3

m − 1.

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}

χe(G ) = min∑

M∈MyM ,

subject to:(1)

∑M∈Me

χe(G ) = min∑

M∈MyM ,

subject to:(1)

∑M∈Me

χe(G ) = min∑

M∈MyM ,

subject to:(1)

∑M∈Me

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.

χ∗e(G ) = min∑

M∈MyM ,

subject to:(1)

∑M∈Me

χ∗e(G ) = max{∆, maxH⊆G ,|V (H)|≥2

|E (H)|b1

2 |V (H)|c}.

If χ∗e(G ) = ∆, then w(G ) ≤ ∆

χ∗e(G ) = min∑

M∈MyM ,

subject to:(1)

∑M∈Me

χ∗e(G ) = max{∆, maxH⊆G ,|V (H)|≥2

|E (H)|b1

2 |V (H)|c}.

If χ∗e(G ) = ∆, then w(G ) ≤ ∆

χ∗e(G ) = min∑

M∈MyM ,

subject to:(1)

∑M∈Me

χ∗e(G ) = max{∆, maxH⊆G ,|V (H)|≥2

|E (H)|b1

2 |V (H)|c}.

If χ∗e(G ) = ∆, then w(G ) ≤ ∆

χ∗e(G ) = min∑

M∈MyM ,

subject to:(1)

∑M∈Me

χ∗e(G ) = max{∆, maxH⊆G ,|V (H)|≥2

|E (H)|b1

2 |V (H)|c}.

If χ∗e(G ) = ∆, then w(G ) ≤ ∆

χ∗e(G ) = min∑

M∈MyM ,

subject to:(1)

∑M∈Me

χ∗e(G ) = max{∆, maxH⊆G ,|V (H)|≥2

|E (H)|b1

2 |V (H)|c}.

If χ∗e(G ) = ∆, then w(G ) ≤ ∆

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 )→∞.

Progress toward Goldberg’s Conjecture

Goldberg’s conjecture has been verified for the following cases:

(Goldberg, 1974; Anderson, 1977) χe(G ) > 54 ∆ + 2

( Anderson, 1977) χe(G ) > 76 ∆ + 4

(Goldberg, 1984) χe(G ) > 98 ∆ + 6

(Nishizeki, Kashiwagi, 1990; Tashkinov, 2000) χe(G ) > 1110 ∆ + 8

(Favrholdt, Stiebitz, Toft, 2006) χe(G ) > 1312 ∆ + 10

(Scheide, Stiebitz, 2009) χe(G ) > 1514 ∆ + 12

(Chen, Yu, Zang, 2011; Schied, 2010) χe(G ) > ∆(G ) +√

∆2 .