Progress on Albertson's Conjecture

TL;DR

This paper advances the verification of Albertson's conjecture for graphs with chromatic number up to 24, refines bounds on potential counterexamples, and extends these bounds for very large r, significantly narrowing the search for counterexamples.

Contribution

The paper verifies Albertson's conjecture for r ≤ 24 and improves bounds on the size of minimal counterexamples, especially for very large r.

Findings

Confirmed the conjecture for r ≤ 24.

Restricted possible counterexamples for r=25,26.

Extended bounds on the order of minimal counterexamples for large r.

Abstract

Albertson conjectured that every graph with chromatic number $r$ has crossing number at least the crossing number of the complete graph $K_r$. This conjecture was proved for $r\le 12$ by Albertson, Cranston, and Fox; for $r\le 16$ by Bar\'{a}t and T\'{o}th; and for $r\le 18$ by Ackerman. Here we verify it for $r\le 24$; we also greatly restrict the possibilities for counterexamples when $r\in\{25,26\}$. In addition, we strengthen earlier work bounding the order of a minimum counterexample for each choice of $r$: we exclude the possibility that $|G|\ge 2.82r$ and exclude the possibility that $1.228r\le |G|\le 1.768r$. Finally, as $r$ grows, we extend the lower end of this range of excluded orders for a minimum counterexample. In particular: if $r\ge 125{,}000$, then we exclude the possibility that $1.10r\le |G|\le 1.768r$; and if $r\ge 825{,}000$, then we exclude the possibility that $1.05r\le |G|\le 1.768r$.