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Show that the only regular, self-complementary, planar graph is the 5-cycle. Pay special attention to the 4-regular case.

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This looks like homework. Please consider tagging it as such. –  Aryabhata Oct 24 '10 at 14:06
It is homework besides the emphasis on the 4-regular case which is really all I'm interested in here. –  everybodyelse Oct 24 '10 at 14:57

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Here's something to get you started. Let G be such a graph, with n vertices. If G is k-regular, then its complement is n-1-k-regular, so to be self-complementary, we must have k = n-1-k, so k = (n-1)/2. Now try finding a reason why the graph is nonplanar if k > 2.

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Yeah that's the idea. But how for the 4-regular case? –  everybodyelse Oct 24 '10 at 14:58

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