   Chapter 10.4, Problem 5ES ### 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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# Extend the argument given in the proof of Lemma 10.4.1 to show that a tree with more than one vertex has at least two vertices of degree 1.

To determine

To prove:

A tree with more than one vertex has at least two vertices of degree 1.

Explanation

Given:

The lemma, any tree that has more than one vertex has at least one vertex of degree 1.

Proof:

Let T be a tree with more than 1 vertex.

Let v0 be a vertex of T and let e0 be an edge incident on v. Such an edge e needs to exist, because T contains more than 1 vertex and a tree cannot contain any isolated vertices (as a tree needs to be connected).

While deg(v0)>1, choose an edge e that is incident on v such that ee (which exists as deg(v0)>1 ) and let v be the vertex on the other end of the edge e (which exists as a tree contains no loops).

Let then v0=v and e0=e.

Repeat until you obtain that deg(v0)=1 (which needs to occur as the tree is finite).

v0 is then, first vertex of degree 1

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