The Minor Crossing Number of Graphs with an Excluded Minor

TL;DR

This paper establishes an upper bound on the minor crossing number for graphs excluding a fixed minor, showing it is linearly bounded by the number of vertices, which advances understanding of graph crossing properties.

Contribution

It proves that for any fixed graph H, graphs excluding H as a minor have a bounded minor crossing number proportional to their size, a new universal bound.

Findings

Minor crossing number is linearly bounded by vertices for H-minor-free graphs.

Existence of a universal constant c for each H.

Extends crossing number theory to minor-closed graph classes.

Abstract

The "minor crossing number" of a graph $G$ is the minimum crossing number of a graph that contains $G$ as a minor. It is proved that for every graph $H$ there is a constant $c$, such that every graph $G$ with no $H$-minor has minor crossing number at most $c|V(G)|$.