Hyperbolicity theorems for correspondence colouring

Abstract

We generalize a framework of list colouring results to correspondence colouring. Correspondence colouring is a generalization of list colouring wherein we localize the meaning of the colours available to each vertex. As pointed out by Dvořák and Postle, both of Thomassen's theorems on the 5-choosability of planar graphs and 3-choosability of planar graphs of girth at least five carry over to the correspondence colouring setting. In this paper, we show that the family of graphs that are critical for 5-correspondence colouring as well as the family of graphs of girth at least five that are critical for 3-correspondence colouring form hyperbolic families. Analogous results for list colouring were shown by Postle and Thomas and by Dvořák and Kawarabayashi, respectively. Using results on hyperbolic families due to Postle and Thomas, we show that this implies that there exists a universal constant c such that if Σ is a surface of Euler genus g, every graph of edge-width at least $c\cdot \log (g+1)$ embedded in Σ is 5-correspondence colourable. This is asymptotically best possible, and improves upon the best known edge-width bound due to Kim, Kostochka, Li, and Zhu. Using results of Dvořák and Kawarabayashi, we show further that there exist linear-time algorithms for the decidability of 5-correspondence colouring for embedded graphs. We show analogous results for 3-correspondence colouring graphs of girth at least five.