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

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

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.

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.

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

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)

Theorem 1 In any graph, the sum of the degrees of all vertices is equal to twice the number of edges.

Corollary In any graph, the number of vertices of odd degree is even.

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.

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?

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.

Theorem 2 A graph G is bipartite if and only if every circuit in G has even length.

Example 5: Testing for a Bipartite Graph Is the graph on the left bipartite?