On 13-crossing-critical graphs with arbitrarily large degrees

Abstract

A recent result of Bokal et al. (2022) [3] proved that the exact minimum value of c such that c-crossing-critical graphs do not have bounded maximum degree is $c=13$. The key to that result is an inductive construction of a family of 13-crossing-critical graphs with many vertices of arbitrarily high degrees. While the inductive part of the construction is rather easy, it all relies on the fact that a certain 17-vertex base graph has the crossing number 13, which was originally verified only by a machine-readable computer proof. We provide a relatively short self-contained computer-free proof of the latter fact. Furthermore, we subsequently generalize the critical construction in order to provide a definitive answer to another long-standing question of this research area; we prove that for every $c\ge 13$ and integers $d,q$, there exists a c-crossing-critical graph with more than q vertices of each of the degrees $3,4,\dots ,d$.