   Chapter 10.1, Problem 39ES ### 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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# Give two examples of graphs that have Hamiltonian circuits but not Euler circuits.

To determine

Give two examples of graphs that have Hamiltonian circuits but not Euler circuits.

Explanation

Given information:

Graphs that have Hamiltonian circuits but not Euler circuits.

Calculation:

We know that:

The degree of a vertex is the number of edges that connect to the vertex.

Note: a loop at a vertex counts as two edges.

An Euler circuit is a circuit that contains edge of the graph.

A Hamiltonian circuit is a simple circuit that passes through every vertex exactly once.

A connected graph has an Euler circuit if and only if each of the vertices has an even degree.

Two examples of graphs with Hamilton circuits but no Euler circuits are given in the image below.

The two graphs below do not contain an Euler circuit, because they each contain at least one vertex with an odd degree. All four vertices in the first example have degree 3, while vertex h and e in the second example have degree 3...

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