The Crossing Number of Seq-Shellable Drawings of Complete Graphs

TL;DR

This paper introduces seq-shellable drawings of complete graphs and proves the Harary-Hill conjecture for this new class, expanding the known classes for which the conjecture holds.

Contribution

The paper defines seq-shellable drawings and demonstrates that it strictly contains bishellable drawings, verifying the Harary-Hill conjecture for this broader class.

Findings

Seq-shellable drawings include all bishellable drawings.

The Harary-Hill conjecture is verified for seq-shellable drawings.

A non-bishellable but seq-shellable drawing of K11 is constructed.

Abstract

The Harary-Hill conjecture states that for every $n>0$ the complete graph on $n$ vertices $K_n$, the minimum number of crossings over all its possible drawings equals \begin{align*} H(n) := \frac{1}{4}\Big\lfloor\frac{n}{2}\Big\rfloor\Big\lfloor\frac{n-1}{2}\Big\rfloor\Big\lfloor\frac{n-2}{2}\Big\rfloor\Big\lfloor\frac{n-3}{2}\Big\rfloor\text{.} \end{align*} So far, the lower bound of the conjecture could only be verified for arbitrary drawings of $K_n$ with $n\leq 12$. In recent years, progress has been made in verifying the conjecture for certain classes of drawings, for example $2$-page-book, $x$-monotone, $x$-bounded, shellable and bishellable drawings. Up to now, the class of bishellable drawings was the broadest class for which the Harary-Hill conjecture has been verified, as it contains all beforehand mentioned classes. In this work, we introduce the class of seq-shellable drawings and verify the Harary-Hill conjecture for this new class. We show that bishellability implies seq-shellability and exhibit a non-bishellable but seq-shellable drawing of $K_{11}$, therefore the class of seq-shellable drawings strictly contains the class of bishellable drawings.