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I know that in the complete bipartite graph $K_{n,n}$ , there is $\frac{n!(n-1)!}{2}$ or $n!(n-1)!$ Hamilton cycles. wiki says first, wolfram says the second one. I know that there is $2n$ ways to specify the "start", but why it goes like $n!(n-1)!$ ?

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    $\begingroup$ Consider small examples like $K_{3,3}$ and count for yourself. $\endgroup$ – Moritz Nov 28 '15 at 10:53
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    $\begingroup$ Whether you divide by $2$ or not depends on whether you consider a cycle to be the same if you reverse its direction. $\endgroup$ – hmakholm left over Monica Nov 28 '15 at 11:01
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As the graph is the complete bipartite graph, we can count the number of cycle as :

  1. Choose an initial set
  2. On the first set, you have $n$ choices for the first vertex
  3. On the second again $n$ choices
  4. Then $n-1$ choices
  5. and so on $\ldots$

Therefore we count H=2(n!)(n!) Hamiltonian cycles. However, we count each cycles $2n$ times because for any cycle there are $2n$ possibles vertices acting as "start". therefore we have $$H = \frac{2(n!)^2}{2n}=n!(n-1)!$$

Now, if you consider a cycle and its reverse as the same cycle, we you should divide this result by 2.

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