$(3a:a)$-List-Colorability of Embedded Graphs of Girth at Least Five

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.
Comment: 32 pages, no figures; updated for reviewer comments, added a more detailed hyperbolicity argument as an appendix