Tight Upper Bounds on the Crossing Number in a Minor-Closed Class

TL;DR

This paper establishes tight upper bounds on the crossing number for graphs excluding a fixed minor, showing it is proportional to the product of maximum degree and number of vertices, improving previous bounds.

Contribution

The paper proves the crossing number for minor-closed graph classes is $O( ext{max degree} imes n)$, resolving an open problem and extending bounds to convex and rectilinear crossing numbers.

Findings

Crossing number is $O( ext{max degree} imes n)$ for minor-free graphs.

Convex crossing number of bounded pathwidth graphs is $O( ext{max degree} imes n)$.

Rectilinear crossing number of $K_{3,3}$-minor-free graphs is bounded by sum of squared degrees.

Abstract

The crossing number of a graph is the minimum number of crossings in a drawing of the graph in the plane. Our main result is that every graph $G$ that does not contain a fixed graph as a minor has crossing number $O(\Delta n)$, where $G$ has $n$ vertices and maximum degree $\Delta$. This dependence on $n$ and $\Delta$ is best possible. This result answers an open question of Wood and Telle [New York J. Mathematics, 2007], who proved the best previous bound of $O(\Delta^2 n)$. We also study the convex and rectilinear crossing numbers, and prove an $O(\Delta n)$ bound for the convex crossing number of bounded pathwidth graphs, and a $\sum_v\deg(v)^2$ bound for the rectilinear crossing number of $K_{3,3}$-minor-free graphs.