Shellable drawings and the cylindrical crossing number of $K_n$

The Harary-Hill Conjecture states that the number of crossings in any drawing of the complete graph Kn in the plane is at least Z(n) := 1 4 ⌊ n 2 ⌋ ⌊ n−1 2 ⌋ ⌊ n−2 2 ⌋ ⌊ n−3 2 ⌋ . In this paper, we settle the Harary-Hill conjecture for shellable drawings. We say that a drawing D of Kn is s-shellable if there exist a subset S = {v1, v2, . . . , vs} of the vertices and a region R of D with the following property: For all 1 ≤ i < j ≤ s, if Dij is the drawing obtained from D by removing v1, v2, . . . vi−1, vj+1, . . . , vs, then vi and vj are on the boundary of the region of Dij that contains R. For s ≥ n/2, we prove that the number of crossings of any s-shellable drawing of Kn is at least the long-conjectured value Z(n). Furthermore, we prove that all cylindrical, x-bounded, monotone, and 2-page drawings of Kn are s-shellable for some s ≥ 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.