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MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 3, Friday, September 5

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Complete graph K n. 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 K n. K 20

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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 K n.

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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 of Directed 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 23 (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 class meeting. Please always bring your updated list of terms to 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 K n is n-1. There are n vertices. The total sum is n(n- 1) = twice the number of edges. K n has n(n-1)/2 edges. On the left K 15 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 V L and V R and every edge joins a vertex in V L with a vertex in V R 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?

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