On 13-Crossing-Critical Graphs with Arbitrarily Large Degrees

TL;DR

This paper offers a computer-free proof that a specific 17-vertex graph has crossing number 13 and generalizes the construction to create c-crossing-critical graphs with high degrees, answering open questions in graph theory.

Contribution

Provides a self-contained, computer-free proof of the crossing number of a key graph and extends the construction to all c≥13, with arbitrary degree distributions.

Findings

Confirmed the crossing number of a 17-vertex graph without computer assistance.

Constructed c-crossing-critical graphs with vertices of arbitrarily high degrees.

Answered an open question by generalizing the critical graph construction.

Abstract

A recent result of Bokal et al. [Combinatorica, 2022] 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 a remaining open question of this research area; we prove that for every c>=13 and integers d,q, there exists a c-crossing-critical graph with more than q vertices of each of the degrees 3,4,...,d.