## Thinking Mathematically (6th Edition)

Published by Pearson

# Chapter 14 - Graph Theory - 14.2 Euler Paths and Euler Circuits - Exercise Set 14.2: 56

#### Answer

The graph has exactly two odd vertices. Therefore, by Euler's theorem, the graph has at least one Euler path. Since this graph has an Euler path, it is possible to find a path that crosses each common state border exactly once.

#### Work Step by Step

According to Euler's theorem, for a graph to have at least one Euler path, the number of odd vertices must be either 0 or 2. Vertex ME and vertex NH are odd vertices. The other vertices are even vertices. The graph has exactly two odd vertices. Therefore, by Euler's theorem, the graph has at least one Euler path. An Euler path is a path that travels through each edge on the graph exactly once. Since this graph has an Euler path, it is possible to find a path that crosses each common state border exactly once.

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