Amini, Omid, Esperet, Louis and van den Heuvel, Jan
(2013)
*A unified approach to distance-two colouring of graphs on surfaces.*
Combinatorica, 33
(3).
pp. 253-296.
ISSN 0209-9683

## 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 {Mathematical expression} plus a constant.

Item Type: | Article |
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Official URL: | http://www.springer.com/new+%26+forthcoming+titles... |

Additional Information: | © 2013 János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg |

Library of Congress subject classification: | Q Science > QA Mathematics |

Sets: | Departments > Mathematics |

Date Deposited: | 17 Jul 2013 07:35 |

URL: | http://eprints.lse.ac.uk/51118/ |

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