Graph Theory Closed Trail

[tarheelborn]

Homework Statement

All vertices in a closed trail have even degree.

Homework Equations

The Attempt at a Solution

Intuitively, I know this statement is true, but I can't seem to see a clear way to show it. I know that a closed trail is a path that connects vertices, so one would follow an edge through a vertex to another edge, thus indicating that, on this particular path, this particular vertex has degree two. Clearly, a non-closed trail has endpoints which must have odd degree because they reach a stopping point on the trail, with a vertex incident to only one edge. Will you please help me put these facts into a coherent proof? Thank you.

 

[verty]

Hint: how could you make a closed trail shorter? Also, be sure you know exactly what a closed trail is.

 

[tarheelborn]

We could make a closed trail shorter by deleting a vertex. If we delete a vertex, the edges incident to that vertex are also deleted, so the trail would be shorter. But this action wouldn't necessarily affect the degree of vertices, would it?