Shellable Drawings and the Cylindrical Crossing Number of $$K_{n}$$ K n

Abstract

The Harary---Hill Conjecture states that the number of crossings in any drawing of the complete graph $$K_n$$Kn in the plane is at least $$Z(n):=\frac{1}{4}\left\lfloor \frac{n}{2}\right\rfloor \left\lfloor \frac{n-1}{2}\right\rfloor \left\lfloor \frac{n-2}{2}\right\rfloor \left\lfloor \frac{n-3}{2}\right\rfloor $$Z(n):=14n2n-12n-22n-32. In this paper, we settle the Harary---Hill conjecture for shellable drawings. We say that a drawing $$D$$D of $$K_n$$Kn is $$s$$s-shellable if there exist a subset $$S = \{v_1,v_2,\ldots , v_ s\}$$S={v1,v2,?,vs} of the vertices and a region $$R$$R of $$D$$D with the following property: For all $$1 \le i < j \le s$$1≤i