# Characterizing graphs with crossing number at least 2

Published:

Recommended citation: Arroyo, A. and Richter, R. B. (2017), Characterizing Graphs with Crossing Number at Least 2. J. Graph Theory, 85: 738-746. doi:10.1002/jgt.22102 https://arxiv.org/pdf/1804.09317.pdf

Our main result includes the following, slightly surprising, fact: a $4$‐connected nonplanar graph $G$ has crossing number at least 2 if and only if, for every pair $\{e,f\}$ of edges having no common incident vertex, there are vertex‐disjoint cycles in $G$ with one containing $e$ and the other containing $f$.