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Graph Theory Closed Trail

  1. Sep 5, 2013 #1
    1. The problem statement, all variables and given/known data

    All vertices in a closed trail have even degree.

    2. Relevant equations



    3. 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.
     
  2. jcsd
  3. Sep 5, 2013 #2

    verty

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    Homework Helper

    Hint: how could you make a closed trail shorter? Also, be sure you know exactly what a closed trail is.
     
  4. Sep 5, 2013 #3
    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?
     
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