How many eulerian cycles are there in a graph with n vertices? The way that I see it there would be $\frac{n!}{(n!)(n-n)!}$ but that simplifies to 1 cycle and I know that there are more cycles than that. What am I missing here? In this case the order of the vertices does not matter.
2026-03-25 22:11:16.1774476676
How many Hamiltonian cycles are there in a graph with n vertices?
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The amount of hamiltonian cycles in a graph is $\frac{1}{2}(n-1)!$ since the graph is un-directed you do not have a choice for the first vertex and every point after that is $(n-1)(n-2)...$ or $(n-1)!$ and since there are 2 ways to choose the same vertex you end up with $\frac{1}{2}(n-1)!$.