CDAM: Computational, Discrete and Applicable Mathematics@LSE |
|
CDAM Research Report, LSE-CDAM-2008-23December 2008 |
A Unified Approach to Distance-Two Colouring of Graphs on Surfaces
Omid Amini, Louis Esperet, and Jan van den Heuvel
In 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.
A PDF file (446 kB) with the full contents of this report can be downloaded by clicking here.
Alternatively, if you would like to get a free hard copy of this report, please send the number of this report, LSE-CDAM-2008-23, together with your name and postal address to:
CDAM Research Reports Series Centre for Discrete and Applicable Mathematics London School of Economics Houghton Street London WC2A 2AE, U.K. |
||
Phone: +44(0)-20-7955 7494. Fax: +44(0)-20-7955 6877. Email: info@maths.lse.ac.uk |
Introduction to the CDAM Research Report Series. | ||
CDAM Homepage. |