   Chapter 10.3, Problem 29ES ### 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 Hamiltonian circuit

To determine

To prove:

Prove that the property (has a Hamiltonian circuit) 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 Hamiltonian circuit.

We then need to show that G has a Hamiltonian circuit.

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 e.g(v) is an endpoint of h(e)

Let k be a positive integer and let the Hamilton circuit in G be  v 1 e 1 v 2 e 2 .... v k e k v 1 with  v i V( G )

and  e i E( G ) gor i=1,2,...,k.

This then implies that the Hamilton circuit contains at least one edge,  e 1 , e 2 ,..., e k  contains no repeated

edges,  v 1 v 2 ,..., v k  contains no repeated vertices and  v 1 , v 2 ,..., v k  are all vertices in G.

g( v 1 )h( e 1 )g( v 2 )h( e 2 )....g( v k )h( e k )g( v 1 ) is then a walk of length k in  G ,because vis an endpoint of e.

g( v ) is an endpoint of h( e ).

Moreover, this walk contains at least one edge (as  v 1 e 1 v 2 e 2

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