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Graph theory: Existence of cycles

  1. Sep 14, 2012 #1
    1. The problem statement, all variables and given/known data
    Let G be a graph containing a cycle C, assume that G contains a path P of length at least k between two verticies on C.
    Show that G contains a cycle of length at least √k.


    3. The attempt at a solution
    Since C is a cycle, there are two paths between a and b. If P intersects none or one of these paths there is no problem.
    If P intersects both there is a problem.

    In the case of one intersection of both paths, there is cycle of length at least k/2 by combining pieces of C and pieces of P. Draw this to see what I mean.

    (No solution for intermediate steps)

    If P intersects P in more than √k places, then C is of length at least √k.
     
  2. jcsd
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