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arXiv · 0812.1345

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

Omid Amini·Louis Esperet·Jan van den Heuvel
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Abstract

In this paper we introduce the notion of $Σ$-colouring of a graph $G$: For given subsets $Σ(v)$ 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 $Σ(v)$ 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 $Δ$ embeddable in some fixed surface is at most $\frac32\,Δ$ plus a constant.

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Omid Amini, Louis Esperet, Jan van den Heuvel. 2012-05-06. A Unified Approach to Distance-Two Colouring of Graphs on Surfaces. https://doi.org/10.1007/s00493-013-2573-2

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