   Chapter 10.1, Problem 49ES ### 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 if there is a circuit in a graph that starts and ends at a vertex v and if w is another vertex in the circuits, then there is a circuit in the graph that starts and ends at w.

To determine

Prove that if there is a circuit in a graph that starts and ends at a vertex v and if w is another vertex in the circuit, then there is a circuit in the graph that starts and ends at w.

Explanation

Given information:

There is a circuit in a graph that starts and ends at a vertex v.

Proof:

Let ve1v1e2v2...vn1enwen+1vn+1en+2vn+2...vn+m1en+mv be a circuit of the graph G that starts and ends at some vertex v and that contains some other vertex w (for some positive integer n and some positive integer m ).

Note that we assume that v and w are distinct.

Since ve1v1e2v2...vn1enwen+1vn+1en+2vn+2...vn+m1en+mv is a circuit, ve1v1e2v2...vn1enwen+1vn+1en+2vn+2...vn+m1en+mv is a closed walk with at least one edge and no repeated edges.

wen+1vn+1en+2vn+2...vn+m1en+mve1v1e2v2

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