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Hey taking the complement of the complete bipartite graph $K_{m,n}$ I think that I get a disconnected graph composed of the complete graph $K_m$ and the complete graph $K_n$ is that right?

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Yes, that is correct. The $m$ vertices are not connected to each other in $K_{m,n}$, so $K_m$ forms in the complement, and along similar lines of reasoning, $K_n$ is in the complement also. However they are not connected since the $m$ vertices and $n$ vertices are all connected to each other.

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    $\begingroup$ Thanks for the help and for editing the question. $\endgroup$ – Benjamin Pallay Oct 5 '14 at 7:33

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