CDAM Research Report, LSE-CDAM-2008-23
A Unified Approach to Distance-Two Colouring of Graphs on Surfaces
Omid Amini, Louis Esperet, and Jan van den HeuvelIn this paper we introduce the notion of (A,B)-colouring of a graph: For given vertex sets A,B, this is a colouring of the vertices in B so that both adjacent vertices and vertices with a common neighbour in A 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 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 embedded in a 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 3Δ⁄2 plus a constant.
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