The Crossing Number of Semi-Pair-Shellable Drawings of Complete Graphs

TL;DR

This paper proves the Harary-Hill Conjecture for a new class of graph drawings called semi-pair-shellable, using novel techniques that relax previous restrictions on reference faces and introduce the concept of k-deviations.

Contribution

It introduces semi-pair-shellable drawings and proves the Harary-Hill Conjecture for this class with new methods involving k-edges and k-deviations.

Findings

Proved the Harary-Hill Conjecture for semi-pair-shellable drawings.

Developed new techniques to relax fixed reference face restrictions.

Introduced the concept of k-deviations to analyze k-edges.

Abstract

The Harary-Hill Conjecture states that for $n\geq 3$ every drawing of $K_n$ has at least \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 \end{align*} crossings. In general the problem remains unsolved, however there has been some success in proving the conjecture for restricted classes of drawings. The most recent and most general of these classes is seq-shellability. In this work, we improve these results and introduce the new class of semi-pair-shellable drawings. We prove the Harary-Hill Conjecture for this new class using novel results on $k$-edges. So far, approaches for proving the Harary-Hill Conjecture for specific classes rely on a fixed reference face. We successfully apply new techniques in order to loosen this restriction, which enables us to select different reference faces when considering subdrawings. Furthermore, we introduce the notion of $k$-deviations as the difference between an optimal and the actual number of $k$-edges. Using $k$-deviations, we gain interesting insights into the essence of $k$-edges, and we further relax the necessity of fixed reference faces.