(a) The complement of a graph is defined as having the same vertex set as the graph, with vertices being adjacent in the complement if and only if they are not adjacent in the graph.

Show that no planar graph of order greater than 10 has a planar complement.

What is the maximum order of a bipartite graph that has a bipartite complement?

(b) For the remainder of this question, let $G$ be a connected bridgeless planar graph with $n \geqslant 4$ vertices, $e$ edges, and containing no circuit of length 4 . Suppose that it is drawn with $f$ faces, of which $t$ are 3-sided.

Show that $2 e \geqslant 3 t+5(f-t)$. Show further that $e \geqslant 3 t$, and hence $f \leqslant 8 e / 15$.

Deduce that $e \leqslant 15(n-2) / 7$. Is there some $n$ and some $G$ for which equality holds? [Hint: consider "slicing the corners off" a dodecahedron.]