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Chapter 14: Problem 64

How do you determine if a graph has at least one Euler circuit?

Short Answer

Expert verified
A graph has at least one Euler circuit if it is connected and every one of its vertices has an even degree.

Step by step solution

01

Definition of Euler Circuit

An Euler circuit in a graph is a path that visits every edge exactly once and returns to its starting vertex. It's named after the Swiss mathematician Leonhard Euler who first proposed the concept.
02

Conditions for an Euler Circuit

In order for a graph to have an Euler circuit, it must satisfy two conditions: The graph must be connected, meaning there's a path between every pair of vertices in the graph. Secondly, every vertex of the graph has to have an even degree. The degree of a vertex is the number of edges that are incident on it.
03

Checking the Conditions

To determine if a graph has an Euler circuit: 1. Check if the graph is connected. If it's not, it can't have an Euler circuit. 2. Check the degree of every vertex in the graph. If every vertex has an even degree, then the graph has at least one Euler circuit; if any vertex has an odd degree, the graph does not have an Euler circuit.

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Most popular questions from this chapter

What are adjacent vertices? If two vertices are near each other in a graph, are they necessarily adjacent? Explain your answer.

In Exercises 15-18, determine the number of Hamilton circuits in a complete graph with the given number of vertices. 4

Why is the Brute Force Method impractical for large numbers of vertices?

In Exercises 11-16, a graph with no loops or more than one edge between any two vertices is described. Which one of the following applies to the description? i. The described graph is a tree. ii. The described graph is not a tree. iii. The described graph may or may not be a tree. The graph has nine vertices and six edges.

Describe the relationship between the number of vertices and the number of edges in a tree.
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