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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 KnKn in the plane is at least Z(n):=14⌊n2⌋⌊n−12⌋⌊n−22⌋⌊n−32⌋Z(n):=14⌊n2⌋⌊n−12⌋⌊n−22⌋⌊n−32⌋. In this paper, we settle the Harary–Hill conjecture for shellable drawings. We say that a drawing DD of KnKn is ss-shellable if there exist a subset S={v1,v2,…,vs}S={v1,v2,…,vs} of the vertices and a region RR of DD with the following property: For all 1≤i<j≤s1≤i<j≤s, if DijDij is the drawing obtained from DD by removing v1,v2,…,vi−1,vj+1,…,vsv1,v2,…,vi−1,vj+1,…,vs, then vivi and vjvj are on the boundary of the region of DijDij that contains RR. For s≥⌊n/2⌋s≥⌊n/2⌋, we prove that the number of crossings of any ss-shellable drawing of KnKn is at least the long-conjectured value Z(n)Z(n). Furthermore, we prove that all cylindrical, xx-bounded, monotone, and 2-page drawings of KnKn are ss-shellable for some s≥n/2s≥n/2 and thus they all have at least Z(n)Z(n) crossings. The techniques developed provide a unified proof of the Harary–Hill conjecture for these classes of drawings.

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