(3a:a)-list-colorability of embedded graphs of girth at least five

TL;DR

This paper investigates the list (3a:a)-colorability of embedded graphs with girth at least five, showing that minimal obstructions form a strongly hyperbolic family, leading to bounds on non-colorable subgraphs.

Contribution

It establishes the hyperbolic nature of minimal obstructions for list (3a:a)-colorability in graphs of girth at least five, providing bounds related to Euler genus.

Findings

Minimal obstructions form a strongly hyperbolic family.

Non-colorable graphs contain subgraphs with O(g) vertices.

Results apply to graphs embedded on surfaces with genus g.

Abstract

A graph G is list (b:a)-colorable if for every assignment of lists of size b to vertices of G, there exists a choice of an a-element subset of the list at each vertex such that the subsets chosen at adjacent vertices are disjoint. We prove that for every positive integer a, the family of minimal obstructions of girth at least five to list (3a:a)-colorability is strongly hyperbolic, in the sense of the hyperbolicity theory developed by Postle and Thomas. This has a number of consequences, e.g., that if a graph of girth at least five and Euler genus g is not list (3a:a)-colorable, then G contains a subgraph with O(g) vertices which is not list (3a:a) colorable.