Crossing numbers of dense graphs on surfaces

TL;DR

This paper establishes tight bounds on the crossing numbers of dense graphs on surfaces, disproving a longstanding conjecture and providing explicit constructions with controlled crossings.

Contribution

It provides the first tight bounds on crossing numbers of dense graphs on surfaces, including a disproof of a 1996 conjecture and explicit geometric constructions.

Findings

Lower bounds on crossings match upper bounds up to constants

Disproof of a 1996 conjecture on crossing numbers

Explicit hyperbolic surface constructions with controlled crossings

Abstract

In this paper, we provide upper and lower bounds on the crossing numbers of dense graphs on surfaces, which match up to constant factors. First, we prove that if $G$ is a dense enough graph with $m$ edges and $\Sigma$ is a surface of genus $g$, then any drawing of $G$ on $\Sigma$ incurs at least $\Omega \left(\frac{m^2}{g} \log ^2 g\right)$ crossings. The poly-logarithmic factor in this lower bound is new even in the case of complete graphs and disproves a conjecture of Shahrokhi, Sz\'ekely and Vrt'o from 1996. Then we prove a geometric converse to this lower bound: we provide an explicit family of hyperbolic surfaces such that for any graph $G$, sampling the vertices uniformly at random on this surface and connecting them with shortest paths yields $O\left(\frac{m^2}{g} \log ^2 g\right)$ crossings in expectation.