## Thinking Mathematically (6th Edition)

Published by Pearson

# Chapter 14 - Graph Theory - 14.2 Euler Paths and Euler Circuits - Exercise Set 14.2 - Page 910: 21

#### Answer

(a) This connected graph has exactly two odd vertices. Therefore, by Euler's theorem, the graph has at least one Euler path. (b) A,C,E,D,B,C,E,F is an Euler path.

#### Work Step by Step

(a) Vertex A and vertex F are odd vertices. The other vertices are even vertices. This connected graph has exactly two odd vertices. Therefore, by Euler's theorem, the graph has at least one Euler path. (b) If a graph has exactly two odd vertices, then any Euler path starts at one odd vertex and ends at the other odd vertex. Let's start at vertex A. From there, the path must travel to vertex C. From there, let's travel around the rectangle to vertex E, then to vertex D, then to vertex B, and back to vertex C. There are only two edges which have not been used. We can travel to vertex E. and then finally to vertex F. This path is A,C,E,D,B,C,E,F. This path travels through every edge of the graph exactly once, so it is an Euler path.

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