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My lecture notes in Graph Theory states that a graph of order $n$ and with size (= number of edges) $\binom n 2-(n-2)$ is the maximal graph that does not contain a Hamiltonian cycle.

My question now is how to find such a graph? How can I construct a graph of order $n$ such that this is true?

For the proof that no greater size can be achieved there was a similar question on this site: Proving that a graph of a certain size is Hamiltonian

This can be used to prove that no greater size can be achieved, so I am not interested in this part, only in the constructon part.

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Hint: If a vertex has degree 1, it can't be part of a cycle.

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  • $\begingroup$ Thanks, I did not think of that, this explains the term $\binom n 2$, where does $n-2$ come from? $\endgroup$ Nov 9 '13 at 15:45
  • $\begingroup$ @UlrichOtto What is the degree of a vertex in a complete graph with $n$ vertices? $\endgroup$
    – Calvin Lin
    Nov 9 '13 at 15:46
  • $\begingroup$ Degree is $n-1$ $\endgroup$ Nov 9 '13 at 15:47
  • $\begingroup$ Oh ok, I have to exclude this case because complete graph contains Hamiltonian cycle, but there is still n-2 instead of n-1 $\endgroup$ Nov 9 '13 at 15:49
  • $\begingroup$ @UlrichOtto Read my hint ... What is $(n-1) - 1$? $\endgroup$
    – Calvin Lin
    Nov 9 '13 at 15:50

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