[joe_the_user]

I don't think that's the mistake involved.

Rather, he simply (incorrectly as others have pointed out) asserts a 1-1 map between his problem and the Hamiltonian circuit problem. IE: "A Hamiltonian circuit is a path that traverses an entire graph by visting each node exactly one time. It turns out that this is exactly the problem I was trying to solve. " (exactly his problem equals, has a 1-1 map to Hamiltonian circuits).

Edit: his problem doesn't map either way to the Hamiltonian graph problem - you could have two disjoint graphs that could be solved as Secret Santa problems but not as Hamiltonian graph problems.