A unified approach to distance-two colouring of planar graphs

摘要

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 plane graphs. We prove a general result which implies asymptotic versions of Wegner's and Borodin's Conjecture on 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 from the fractional chromatic index.