  In 1958, Hill conjectured that the minimum number of crossings in a drawing
of $K_n$ is exactly $Z(n) = \frac{1}{4} \lfloor\frac{n}{2}\rfloor
\left\lfloor\frac{n-1}{2}\right\rfloor
\left\lfloor\frac{n-2}{2}\right\rfloor\left\lfloor\frac{n-3}{2}\right\rfloor$.
Generalizing the result by \'{A}brego et al. for 2-page book drawings, we prove
this conjecture for plane drawings in which edges are represented by
$x$-monotone curves. In fact, our proof shows that the conjecture remains true
for $x$-monotone drawings of $K_n$ in which adjacent edges may cross an even
number of times, and instead of the crossing number we count the pairs of edges
which cross an odd number of times. We further discuss a generalization of this
result to shellable drawings, a notion introduced by \'{A}brego et al. We also
give a combinatorial characterization of several classes of $x$-monotone
drawings of complete graphs using a small set of forbidden configurations. For
a similar local characterization of shellable drawings, we generalize
Carath\'eodory's theorem to simple drawings of complete graphs.

Combinatorics;Discrete Mathematics