   Chapter 10.3, Problem 23ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# Prove that each of the properties in 21-29 is an invariant for graph isomorphism. Assume that n, m, and k are all nonnegative integers.Has a circuit of length k

To determine

To prove:

Prove that the property (has a circuit of length k ) is an invariant for graph isomorphism.

Explanation

Given information:

Assume that n, m and k are all nonnegative integers.

Proof:

Let G and G be isomorphic graphs and let the graph G contain a circuit of length k (with k a positive integer). We then need to show that G has a circuit of length k as well.

Let V ( G ) be the set of vertices of G and let V(G) be the set of vertices of G.

Let E(G) be the set of edges of G and let E ( G’ ) be the set of edges of G.

Since G and G are isomorphic, there exists two one-to-one correspondences

g:V(G)V(G) and h:E(G)E(G) such that v is an endpoint of eg(v) is an endpoint of h(e).

Let the circuit of length k in G be v1e1v2e2...vkekv1 with viV(G) and eiE(G) for i=1,2,...,k. this thenimplies that the walk contains at least one edge and e1,e2,...,ek contains no repeated edges.

g(v1)h(e1)g(v2)h(e2)..

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