A unified approach to distance-two colouring of planar graphs

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. Our results are based on a strong structural lemma for planar graphs which also implies that the size of a clique in the square of a planar graph of maximum degree ∆ is at most 3 2 ∆ plus a constant.