Sparse String Graphs and Region Intersection Graphs over Minor-Closed Classes have Linear Expansion

TL;DR

This paper proves that sparse string graphs on fixed surfaces and region intersection graphs over minor-closed classes have linear expansion, with applications to graph coloring.

Contribution

It extends linear expansion results to broader classes of sparse graphs over surfaces and minor-closed classes with combinatorial proofs.

Findings

Sparse string graphs on fixed surfaces have linear expansion.

Sparse region intersection graphs over minor-closed classes have linear expansion.

Results have applications to graph coloring.

Abstract

We prove that sparse string graphs in a fixed surface have linear expansion. We extend this result to the more general setting of sparse region intersection graphs over any proper minor-closed class. The proofs are combinatorial and self-contained, and provide bounds that are within a constant factor of optimal. Applications of our results to graph colouring are presented.