  The Harary-Hill Conjecture States that the number of crossings in any drawing
of the complete graph $ K_n $ in the plane is at least
$Z(n):=\frac{1}{4}\left\lfloor \frac{n}{2}\right\rfloor
\left\lfloor\frac{n-1}{2}\right\rfloor \left\lfloor
\frac{n-2}{2}\right\rfloor\left\lfloor \frac{n-3}{2}\right\rfloor$. In this
paper, we settle the Harary-Hill conjecture for {\em shellable drawings}. We
say that a drawing $D$ of $ K_n $ is {\em $ s $-shellable} if there exist a
subset $ S = \{v_1,v_2,\ldots,v_ s\}$ of the vertices and a region $R$ of $D$
with the following property: For all $1 \leq i < j \leq s$, if $D_{ij}$ is the
drawing obtained from $D$ by removing $v_1,v_2,\ldots
v_{i-1},v_{j+1},\ldots,v_{s}$, then $v_i$ and $v_j$ are on the boundary of the
region of $D_{ij}$ that contains $R$. For $ s\geq n/2 $, we prove that the
number of crossings of any $ s $-shellable drawing of $ K_n $ is at least the
long-conjectured value Z(n). Furthermore, we prove that all cylindrical, $ x
$-bounded, monotone, and 2-page drawings of $ K_n $ are $ s $-shellable for
some $ s\geq n/2 $ and thus they all have at least $ Z(n) $ crossings. The
techniques developed provide a unified proof of the Harary-Hill conjecture for
these classes of drawings.
