Shellable Drawings and the Cylindrical Crossing Number of Kn

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 ) : = 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 K n is s - shellable if there exist a subset S = { v 1 , v 2 , … , v s } of the vertices and a region R of D with the following property: For all 1 ≤ i < j ≤ s , if D i j is the drawing obtained from D by removing v 1 , v 2 , … , v i - 1 , v j + 1 , … , v s , then v i and v j are on the boundary of the region of D i j that contains R . For s ≥ ⌊ 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 ≥ 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.