## Thinking Mathematically (6th Edition)

Published by Pearson

# Chapter 14 - Graph Theory - 14.3 Hamilton Paths and Hamilton Circuits - Exercise Set 14.3: 4

#### Answer

The path A,G,B,C,F,E,D,A is a Hamilton circuit that begins as A,G,...

#### Work Step by Step

A Hamilton path is a sequence of adjacent vertices that visits each vertex in the graph exactly once. A Hamilton circuit is a Hamilton path that begins and ends at the same vertex. Let's start at vertex A. The path can travel to vertex G, then to vertex B, then to vertex C, then to vertex F. After this step, the path must travel to vertex E, then to vertex D, and then finally to vertex A. This path is A,G,B,C,F,E,D,A. Since this path visits every vertex in the graph exactly once (except for vertex A which is the starting and ending vertex), this path is a Hamilton path. Since this Hamilton path starts and ends at the same vertex, this path is a Hamilton circuit. This is one Hamilton circuit that begins as A,G,..., but there are other such Hamilton circuits in the graph.

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