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Depending on your definition of simpler, everyone could just pair off and shake exactly one hand.

It's the simplest option when n is even (which it is in this case), but fails when n is odd.

The article about how this problem is trivial for N even but impossible for N odd.

The article is talking about the biggest odd subgraph within the even subgraph solution.

I think the problem with the suggestion is that the solution to the even subgraph may not be under your control? Not sure.