MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 3, Friday, September 5
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Transcript of MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 3, Friday, September 5
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MATH 310, FALL 2003(Combinatorial Problem
Solving)Lecture 3, Friday, September 5
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Complete graph Kn.
A graph on n vertices in which each vertex is adjacent to all other vertices is called a complete graph on n vertices, denoted by Kn.
K20
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Some complete graphs Here are some
complete graphs. For each one
determine the number of vertices, edges, and the degree of each vertex.
Every graph on n vertices is a subgraph of Kn.
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Example 2: Isomorphism in Symmetric Graphs
The two graphs on the left are isomorphic.
Top graph vertices clockwise: a,b,c,d,e,f,g
Bottom graph vertices clockwise: 1,2,3,4,5,6,7
Possible isomorphism:a-1,b-5,c-2,d-6,e-3,f-7,g-4.
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Example 3: Isomorphism ofDirected Graphs
Some hints how to prove non-isomorphism:
If two graphs are not isomorphic as undirected graphs, they cannot be isomorphic as directed graphs.
(p,q) –label on a vertex: indegree p, outdegree q.
Look at the directed edges and their (p,q,r,s) labels!
1
2 3(p,q,r,s)
(r,s)
(2,3)
(p,q)
e
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1.3. Edge Counting Homework (MATH 310#1F):
• Read 1.4. Write down a list of all newly introduced terms (printed in boldface)
• Do Exercises1.3: 4,6,8,12,13• Volunteers:
• ____________• ____________• Problem: 13.
News: News: Please always bring your updated list of terms to Please always bring your updated list of terms to
class meeting. class meeting. Homework in now labeled for easier identification:Homework in now labeled for easier identification:
• (MATH 310, #, Day-MWF)(MATH 310, #, Day-MWF)
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Theorem 1 In any graph, the sum of the
degrees of all vertices is equal to twice the number of edges.
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Corollary In any graph, the number of vertices
of odd degree is even.
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Example 2: Edges in a Complete Graph
The degree of each vertex of Kn is n-1. There are n vertices. The total sum is n(n-1) = twice the number of edges.
Kn has n(n-1)/2 edges.
On the left K15 has 105 edges.
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Example 3: Impossible graph Is it possible to have a group of
seven people such that each person knows exactly three other people in the group?
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Bipartite Graphs
A graph G is bipartite if its vertices can be partitioned into two sets VL and VR and every edge joins a vertex in VL with a vertex in VR
Graph on the left is biparite.
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Theorem 2 A graph G is bipartite if and only if
every circuit in G has even length.
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Example 5: Testing for a Bipartite Graph
Is the graph on the left bipartite?