A Unified Approach to Distance-Two Colouring of Graphs on Surfaces

TL;DR

This paper introduces a generalized graph colouring concept called $\\Sigma$-colouring, proving asymptotic bounds for graphs on surfaces that unify and extend results related to graph squares and cyclic colourings.

Contribution

It presents a unified framework for distance-two colourings on surface-embeddable graphs, linking structural properties to colouring bounds and extending classical conjectures.

Findings

Asymptotic bounds for $\\Sigma$-colouring on fixed surfaces.

Clique size in graph squares is at most 1.5 times maximum degree plus a constant.

Reduction of colouring problems to edge-colouring of multigraphs using recent methods.

Abstract

In this paper we introduce the notion of $\Sigma$-colouring of a graph $G$: For given subsets $\Sigma(v)$ of neighbours of $v$, for every $v\in V(G)$, this is a proper colouring of the vertices of $G$ such that, in addition, vertices that appear together in some $\Sigma(v)$ receive different colours. This concept generalises the notion of colouring the square of graphs and of cyclic colouring of graphs embedded in a surface. We prove a general result for graphs embeddable in a fixed surface, which implies asymptotic versions of Wegner's and Borodin's Conjecture on the planar version of these two colourings. Using a recent approach of Havet et al., we reduce the problem to edge-colouring of multigraphs, and then use Kahn's result that the list chromatic index is close to the fractional chromatic index. Our results are based on a strong structural lemma for graphs embeddable in a fixed surface, which also implies that the size of a clique in the square of a graph of maximum degree $\Delta$ embeddable in some fixed surface is at most $\frac32\,\Delta$ plus a constant.