   Chapter 10.1, Problem 38ES ### 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 Euler circuits but not Hamiltonian circuits.

To determine

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

Explanation

Given information:

Graphs which has Euler circuits but not Hamiltonian 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 Euler circuits but no Hamiltonian circuits are given in the image below.

The two graphs below have an Euler circuit, because the graphs are connected and each vertex has an even degree. For example, bacbdeb is an Euler circuit in the first example and ghi f glkjg is an Euler circuit in the second example

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