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Chapter 14: Problem 48

What is a Hamilton path? How does this differ from an Euler path?

Short Answer

Expert verified
A Hamilton path in a graph is a path that visits each vertex exactly once, while an Euler path visits every edge exactly once. There is no effective method for determining whether a given general graph has a Hamiltonian path due to its NP-completeness, while the existence of an Euler path can be easily determined by checking if the graph is connected and if the number of vertices with odd degree is either 0 or 2.

Step by step solution

01

Definition of Hamilton Path

A Hamilton path in a graph is a path that visits each vertex exactly once. Unlike a Hamilton cycle, the path does not need to end at the starting vertex. The path is named after the mathematician Sir William Rowan Hamilton who first formulated the 'Hamiltonian path problem'.
02

Definition of Euler Path

An Euler path, on the other hand, is a path in a graph which visits every edge exactly once. This path can begin and end at different vertices. This is named after Leonhard Euler, who solved the Seven Bridges of K枚nigsberg problem and accounted for the conditions when such path exists.
03

Differences Between Hamilton and Euler Paths

The main difference between a Hamilton path and an Euler path lies in whether edges or vertices are visited. A Hamilton path travels through each vertex once, without revisiting any, while an Euler path goes through each edge once. Another difference is, unlike Euler paths, there is no effective method for determining whether a given general graph has a Hamiltonian path due to its NP-completeness. On the other hand, an Euler path exists in a graph if it is connected and the number of vertices with odd degree is either 0 or 2.

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Most popular questions from this chapter

Describe how to determine the number of Hamilton circuits in a complete graph.

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