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The complement of $G$ is the graph with the same vertices as $G$, but whose edges are precisely the non-edges of $G$. (That is, $u$ is adjacent to $v$ in the complement of $G$ if and only $u$ is not adjacent to $v$ in $G$.) Therefore, if $G^c$ is the complement of $G$, then $E(K_{|V (G)|} )$ is the disjoint union of $E(G)$ and $E(G^c )$. Show that if $G$ is a simple planar graph with at least eleven vertices, then the complement of $G$ is not planar.

platty
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Hint: Remember Euler's criterion: if $G$ is a planar graph, then $\epsilon\le 3\nu-6$.