   Chapter 10.1, Problem 52ES ### 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 Corollary 10.1.5.

To determine

Prove Corollary 10.2.5.

Explanation

Given information:

G is a graph

v and w are two distinct vertices of G.

Proof:

First part:

Let ve1v1e2v2...en1vn1enw be an Euler path from v to w. this then implies that all edges in G are e1,e2,...,en1,en, while the Euler path also contains all vertices of G at least once.

Let us define a (new) edge e from w to v and let G=G{e} (which is the graph G with an added edge). ve1v1e2v2...en1vn1enwev is then an Euler circuit in G.

A graph contains an Euler circuit if and only if the graph is connected and all vertices of the graph have a positive even degree. This then implies that G is connected and all vertices of G have an even degree.

G is then connected, because G is G without the edge e (even though we removed an edge between v and w, ve1v1e2v2...en1vn1enw is still a walk from v to w )

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