Nested cycles in large triangulations and crossing-critical graphs

TL;DR

This paper proves that large plane triangulations contain many nested cycles with specific intersection properties and uses this to determine finiteness conditions for certain crossing-critical graphs with high average degree.

Contribution

It introduces a new structural property of large triangulations and applies it to settle a question about the finiteness of crossing-critical graphs with given average degree.

Findings

Large triangulations have nested cycles with specific intersection patterns.

Finiteness of k-crossing-critical graphs with average degree at least six is established.

Characterization of when infinite families of crossing-critical graphs exist based on average degree.

Abstract

We show that every sufficiently large plane triangulation has a large collection of nested cycles that either are pairwise disjoint, or pairwise intersect in exactly one vertex, or pairwise intersect in exactly two vertices. We apply this result to show that for each fixed positive integer $k$, there are only finitely many $k$-crossing-critical simple graphs of average degree at least six. Combined with the recent constructions of crossing-critical graphs given by Bokal, this settles the question of for which numbers $q>0$ there is an infinite family of $k$-crossing-critical simple graphs of average degree $q$.