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Chapter 14: Problem 65

How do you determine if a graph has no Euler paths and no Euler circuits?

Short Answer

Expert verified
A graph has no Euler paths and no Euler circuits if it has more than two vertices that have odd degrees.

Step by step solution

01

Understanding Euler Paths and Euler Circuits

To start, one must understand what an Euler path and Euler circuit mean. An Euler circuit in a graph is a path that starts and ends at the same vertex and includes every edge of the graph exactly once. An Euler path, on the other hand, is a path that uses every edge of a graph exactly once. The path does not necessarily have to start and end at the same vertex.
02

Applying the Degree Rule

The key to analyzing whether a graph has an Euler path or circuit is by checking the vertex degrees - the number of edges connecting to a vertex. If all vertices have an even degree, then the graph has an Euler circuit. Accordingly, if a graph has exactly two vertices of odd degree, it has an Euler path.
03

Determining the Absence of Euler Paths and Euler Circuits

To determine if a graph has no Euler path and no Euler circuit, check if more than two vertices have odd degrees. If this is the case, we can conclude that the graph does not contain any Euler path or circuit.

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Most popular questions from this chapter

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

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Ambassadors from countries \(A, B, C, D, E\), and \(F\) are to be seated around a circular conference table. Friendly relations among the various countries are described as follows: \(A\) has friendly relations with \(B\) and \(F\). \(B\) has friendly relations with \(A, C\), and \(E\). C has friendly relations with \(B, D, E\), and \(F\). \(E\) has friendly relations with \(B, C, D\), and \(F\). All friendly relations are mutual. Using vertices to represent countries and edges to represent friendly relations, draw a graph that models the information given. Then use a Hamilton circuit to devise a seating arrangement around the table so that the ambassadors from \(B\) and \(E\) are seated next to each other, and each ambassador represents a country that has friendly relations with the countries represented by the ambassadors next to him or her.

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