Crossings, colorings, and cliques

TL;DR

This paper proves Albertson's conjecture for chromatic numbers 7 through 12, establishing that graphs with high chromatic number have crossing numbers at least as large as complete graphs of the same size, extending previous results.

Contribution

It extends the proof of Albertson's conjecture to chromatic numbers 7 to 12 using bounds on critical graphs and crossing numbers, beyond the cases previously verified.

Findings

Confirmed Albertson's conjecture for r=7 to 12

Established lower bounds on crossing numbers for graphs with high chromatic number

Connected critical graph edge bounds with crossing number estimates

Abstract

Albertson conjectured that if graph $G$ has chromatic number $r$, then the crossing number of $G$ is at least that of the complete graph $K_r$. This conjecture in the case $r=5$ is equivalent to the four color theorem. It was verified for $r=6$ by Oporowski and Zhao. In this paper, we prove the conjecture for $7 \leq r \leq 12$ using results of Dirac; Gallai; and Kostochka and Stiebitz that give lower bounds on the number of edges in critical graphs, together with lower bounds by Pach et.al. on the crossing number of graphs in terms of the number of edges and vertices.