# Tree-like distance colouring for planar graphs of sufficient girth

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 _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.

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