Tree-Like Distance Colouring for Planar Graphs of Sufficient Girth

Abstract

Given a multigraph G and a positive integer t, the distance-t chromatic index of G is the least number of colours needed for a colouring of the edges so that every pair of distinct edges connected by a path of fewer than t edges must receive different colours. Let π ′ t (d) and τ ′ t (d) be the largest values of this parameter over the class of planar multigraphs and of (simple) trees, respectively, of maximum degree d. We have that π ′ t (d) is at most and at least a non-trivial constant multiple larger than τ ′ t (d). (We conjecture lim sup d→∞ π ′ 2 (d)/τ ′ 2 (d) = 9/4 in particular.) We prove for odd t the existence of a quantity g depending only on t such that the distance-t chromatic index of any planar multigraph of maximum degree d and girth at least g is at most τ ′ t (d) if d is sufficiently large. Such a quantity does not exist for even t. We also show a related, similar phenomenon for distance vertex-colouring.