Crossing-critical graphs with large maximum degree

TL;DR

This paper disproves two longstanding conjectures by constructing crossing-critical graphs with large maximum degree for all crossing numbers at least 171, challenging previous beliefs about their structural properties.

Contribution

It provides the first counterexamples to the conjectures that crossing-critical graphs have bounded maximum degree and bounded bandwidth for all crossing numbers.

Findings

Counterexamples exist for all k ≥ 171

Crossing-critical graphs can have arbitrarily large maximum degree

Disproves previous conjectures about structural bounds

Abstract

A conjecture of Richter and Salazar about graphs that are critical for a fixed crossing number $k$ is that they have bounded bandwidth. A weaker well-known conjecture of Richter is that their maximum degree is bounded in terms of $k$. In this note we disprove these conjectures for every $k\ge 171$, by providing examples of $k$-crossing-critical graphs with arbitrarily large maximum degree.