这是一份warwick华威大学MA138-10的成功案例

Show that if $K_{n}$ or $K_{p, \eta}$ has an even number of edges, then it has an optimal biplanar drawing, which is self-complementary.

Concerning upper bounds for $\operatorname{cr}{2}(G)$, in terms of $m$, we proved in a joint paper with Shahrokhi [21] a general upper bound for the $k$-page book crossing numbers of graphs: $$ \mu{k}(G) \leq \frac{1}{3 k^{2}}\left(1-\frac{1}{2 k}\right) m^{2}+O\left(\frac{m^{2}}{k n}\right),
$$
which together with $\mathrm{cr}{2}(G) \leq \mu{4}(G)$ gives a general upper bound on $\mathrm{cr}{2}(G)$ $$ \mathrm{cr}{2}(G) \leq \frac{7}{384} m^{2}+O\left(\frac{m^{2}}{n}\right) .
$$

MA138-10 COURSE NOTES ：

The girth formula yields
$$
\mathrm{cr}{2}\left(K{p, q}\right) \geq p q-4(p+q-2) .
$$
One can use the counting argument with $H=K_{10,10}, G=K_{p, q}$, and the fact that $\mathrm{cr}{2}\left(K{10,10}\right) \geq 28$ from to obtain:
Theorem 1. For $10 \leq p \leq q$, we have
$$
\mathrm{cr}{2}\left(K{p, q}\right) \geq \frac{p(p-1) q(q-1)}{290} .
$$
For $p \leq 9$ we make a finer analysis of $\operatorname{cr}{2}\left(K{p, q}\right)$.