2018/05/29 by Zdeněk Dvořák, Zdenĕk Dvořák, Dvořák, Zdeněk +2
Computer Science · Mathematics · #05C15 #Advanced Graph Theory Research #Advanced Topology and Set Theory #Combinatorics #Combinatorics (math.CO) #Computer science #Discrete mathematics #Disjoint sets #FOS: Mathematics #G.2.2 #Girth (graph theory) #Graph #Integer (computer science) #Limits and Structures in Graph Theory #Mathematics #Vertex (graph theory) #acm:05C15 #math.CO #msc:05C15
paper · pdf · doi:10.48550/arxiv.1805.11507
32 pages, no figures; updated for reviewer comments, added a more detailed hyperbolicity argument as an appendix
openalex publication_date 2018/05/29 · openalex created_date 2018/06/13 · arxiv created 2019/10/11 · arxiv updated 2019/10/14 · openalex updated_date 2026/08/06
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.