Nested cycles in large triangulations and crossing-critical graphs

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. (C) 2011 Gelasio Salazar, Cesar Hernandez-Velez, Robin Thomas. Published by Elsevier Inc. All rights reserved.