We consider two graph colouring problems in which edges at distance at most
t are given distinct colours, for some fixed positive integer t. We obtain
two upper bounds for the distance-t chromatic index, the least number of
colours necessary for such a colouring. One is a bound of (2-\eps)\Delta^t
for graphs of maximum degree at most Δ, where \eps is some absolute
positive constant independent of t. The other is a bound of O(Δt/logΔ) (as Δ→∞) for graphs of maximum degree at most Δ
and girth at least 2t+1. The first bound is an analogue of Molloy and Reed's
bound on the strong chromatic index. The second bound is tight up to a constant
multiplicative factor, as certified by a class of graphs of girth at least g,
for every fixed g≥3, of arbitrarily large maximum degree Δ, with
distance-t chromatic index at least Ω(Δt/logΔ).Comment: 14 pages, 2 figures; to appear in Combinatorics, Probability and
Computin