Bishellable Drawings Of K-N

The Harary-Hill conjecture, still open after more than 50 years, asserts that the crossing number of the complete graph K-n is H(n) := 1/4 left perpendicular n/2 right perpendicular left perpendicular n-1/2 right perpendicular left perpendicular n-2/2 right perpendicular left perpendicular n-3/2 right perpendicular. Abrego et al. [Discrete Comput. Geom., 52 (2014), pp. 743-753] introduced the notion of shellability of a drawing D of K-n. They proved that if D is s-shellable for some s >= left perpendicular n/2 right perpendicular, then D has at least H(n) crossings. This is the first combinatorial condition on a drawing that guarantees at least H(n) crossings. In this work, we generalize the concept of s-shellability to bishellability, where the former implies the latter in the sense that every s-shellable drawing is, for any b <= s - 2, also b-bishellable. Our main result is that (left perpendicular n/2 right perpendicular-2)-bishellability of a drawing D of K-n also guarantees, with a simpler proof than for s-shellability, that D has at least H(n) crossings. We exhibit a drawing of K-11 that has H(11) crossings, is 3-bishellable, and is not s-shellable for any s >= 5. This shows that we have properly extended the class of drawings for which the Harary-Hill conjecture is proved. Moreover, we provide an infinite family of drawings of K-n that are ( (left perpendicular n/2 right perpendicular-2)-bishellable, but not s-shellable for any s >= left perpendicular n/2 right perpendicular.