Bishellable drawings of $K_n$

TL;DR

This paper introduces the concept of bishellability, extending shellability, and proves that bishellable drawings of complete graphs have at least the conjectured minimum number of crossings, thus broadening the class of graphs for which the Harary-Hill conjecture holds.

Contribution

The paper generalizes shellability to bishellability, providing a simpler proof that bishellable drawings meet the Harary-Hill crossing number conjecture, and demonstrates that bishellability encompasses a larger class of drawings.

Findings

Bishellability guarantees at least H(n) crossings in K_n.

A specific 11-vertex drawing is 3-bishellable but not s-shellable for s≥5.

An infinite family of bishellable but not s-shellable drawings is constructed.

Abstract

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) = \frac 1 4 \left\lfloor\frac{\mathstrut n}{\mathstrut 2}\right\rfloor \left\lfloor\frac{\mathstrut n-1}{\mathstrut 2}\right\rfloor \left\lfloor\frac{\mathstrut n-2}{\mathstrut 2}\right\rfloor \left\lfloor\frac{\mathstrut n-3}{\mathstrut 2}\right \rfloor$. \'Abrego et al. introduced the notion of shellability of a drawing $D$ of $K_n$. They proved that if $D$ is $s$-shellable for some $s\geq\lfloor\frac{n}{2}\rfloor$, 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 $b \leq s-2$, also $b$-bishellable. Our main result is that $(\lfloor \frac{n}{2} \rfloor\!-\!2)$-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 $s\geq5$. 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 $(\lfloor \frac{n}{2} \rfloor\!-\!2)$-bishellable, but not $s$-shellable for any $s\geq\lfloor\frac{n}{2}\rfloor$.