# Chapter 14 - Graph Theory - 14.2 Euler Paths and Euler Circuits - Exercise Set 14.2: 22

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

#### Work Step by Step

(a) Vertex C and vertex D 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 C. From there, let's travel around the triangle to vertex A, then to vertex E, and then back to vertex C. From there, let's travel to vertex D, then to vertex B, then to vertex F, and then back to vertex D. This path is C,A,E,C,D,B,F,D. This path travels through every edge of the graph exactly once, so it is an Euler path.

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.