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Let $G$ and $H$ be isomorphic graphs. Prove that the complements of $G$ and $H$ are isomorphic.

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What does "self-complementary both are isomorphism" mean? – Chris Eagle Oct 19 '12 at 19:33
This is rather difficult to understand. Can you restate this more clearly? – Cameron Buie Oct 19 '12 at 19:38
I edited the question to something that almost makes sense, but am not sure if that resembles the askers intention. Whatever self-complementaries are, they are probably trivially isomorphic for isomorphic inputs. – Hagen von Eitzen Oct 19 '12 at 19:53
sorry for bad english. – geni Oct 20 '12 at 7:23
"prove that the complements of G and H are isomorphic" is true. – geni Oct 20 '12 at 7:24
up vote 5 down vote accepted

I am going to answer the question that I see, which is to "Prove that the complements of $G$ and $H$ are isomorphic." I can't think of any other possible meaning to the question.

You're telling me $G$ and $H$ are isomorphic, so that means there exists a map from the vertices of $G$ to the vertices of $H$ such that $u$ is adjacent to $v$ in $G$ if and only if $f(u)$ is adjacent to $f(v)$ in $H$.

So, now you want to know if the complements of $G$ and $H$ are isomorphic?

Hint 1: If $u$ and $v$ are adjacent in $G$, what is true about $u$ and $v$ in the complement of $G$? Or, if $u$ and $v$ are not adjacent in $G$, what is true about $u$ and $v$ in the complement? Similarly, with $H$.

Hint 2: Use the same $f$ you already know exists since $G$ is isomorphic to $H$.

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