MATH 310 FALL 2003 Combinatorial Problem Solving Lecture

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

Complete graph Kn. n K 20 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.

Some complete graphs n n n 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.

Example 2: Isomorphism in Symmetric Graphs n n 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 n n (2, 3) 1 n (p, q) 2 (p, q, r, s) e (r, s) 3 n 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. 3. Edge Counting n Homework (MATH 310#1 F): • Read 1. 4. Write down a list of all newly introduced terms (printed in boldface) • Do Exercises 1. 3: 4, 6, 8, 12, 13 • Volunteers: • ____________ • Problem: 13. n News: n n Please always bring your updated list of terms to class meeting. Homework in now labeled for easier identification: • (MATH 310, #, Day-MWF)

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

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

Example 2: Edges in a Complete Graph n n n 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 K 15 has 105 edges.

Example 3: Impossible graph n 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 VL and VR and every edge joins a vertex in VL with a vertex in VR Graph on the left is biparite.

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

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