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arXiv · 2303.16997

Hyperbolicity Theorems for Correspondence Colouring

Luke Postle·Evelyne Smith-Roberge
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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 Dvo\v{r}\'ak 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 Dvo\v{r}\'ak and Kawarabayashi, respectively. Using results on hyperbolic families due to Postle and Thomas, we show further that this implies that locally planar graphs are 5-correspondence colourable; and, using results of Dvo\v{r}\'ak and Kawarabayashi, 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.

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Luke Postle, Evelyne Smith-Roberge. 2023-03-29. Hyperbolicity Theorems for Correspondence Colouring. https://arxiv.org/abs/2303.16997

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