Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top


I asked my professor and he said that a counter example would be two nodes, by which the pathw ould go from one node and back. this would be a closed path but does not contain a cycle. But I am confused after looking at this again. Why is this not a cycle? Need there be another node?

share|cite|improve this question
This may depend on your local definitions of walk, path, cycle. – Hagen von Eitzen May 25 '13 at 13:28
A walk that contains each vertex at most once, is called a cycle if tis closed and the start and end point are equal. Path being the same, only open. – WiseStrawberry May 25 '13 at 13:31
up vote 2 down vote accepted

I guess the answer depends on the exact definition of cycle. If it is as you wrote in your comment - a closed walk that starts and ends in the same vertex, and no vertex repeats on the walk (except for the start and end), then your example with two nodes is a cycle.

However, a definition of a cycle usually contains a condition of non-triviality stating that a cycle has at least three vertices. So a graph with two vertices is not a cycle according to this definition.

share|cite|improve this answer
as I suspected, mostly by geometric interpretation of a cycle not being the same as line.. – WiseStrawberry May 25 '13 at 13:59

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.