Bishellable drawings of Kn

The Harary-Hill conjecture, still open after more than 50 years, asserts that the crossing number of the complete graph Kn is H(n) = 1 4 ⌊ n 2 ⌋⌊ n− 1 2 ⌋⌊ n− 2 2 ⌋⌊ n− 3 2 ⌋ . Ábrego et al. [3] introduced the notion of shellability of a drawing D of Kn. They proved that if D is s-shellable for some s ≥ b 2 c, 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 (b 2 c−2)-bishellability also guarantees, with a simpler proof than for s-shellability, that a drawing has at least H(n) crossings. We exhibit a drawing of K11 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.