## 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*∈*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 \( \frac{3} {2}\Delta \) plus a constant.

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## Additional information

This paper benefited greatly from helpful comments of anonymous referees. The authors would like to thank the referees for careful reading of the paper and for their constructive suggestions.

The research for this paper was started during a visit of LE and JvdH to the Mascotte research group at INRIA Sophia-Antipolis, where OA was a PhD student (joint with École Polytechnique). The authors like to thank the members of Mascotte for their hospitality.

JvdH’s visit to INRIA Sophia-Antipolis was partly supported by a grant from the Alliance Programme of the British Council.

Part of this research has been conducted while OA was visiting McGill University in Montreal. He warmly thanks Bruce Reed for providing the possibility for such a visit.

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Amini, O., Esperet, L. & Van Den Heuvel, J. A unified approach to distance-two colouring of graphs on surfaces.
*Combinatorica* **33, **253–296 (2013). https://doi.org/10.1007/s00493-013-2573-2

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### Mathematics Subject Classification (2010)

- 05C15
- 05C10