MATH 310, FALL 2003(Combinatorial Problem

Solving)Lecture 3, Friday, September 5

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

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.

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 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

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)

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 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.

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 VL and VR and every edge joins a vertex in VL with a vertex in VR

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?