## The distance-t chromatic index of graphs

### Abstract

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 $\Delta$, where $\eps$ is some absolute positive constant independent of $t$. The other is a bound of $O(\Delta^t/\log \Delta)$ (as $\Delta\to\infty$) for graphs of maximum degree at most $\Delta$ 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 $g \ge 3$, of arbitrarily large maximum degree $\Delta$, with distance-$t$ chromatic index at least $\Omega(\Delta^t/\log \Delta)$.Comment: 14 pages, 2 figures; to appear in Combinatorics, Probability and Computin

Topics: Mathematics - Combinatorics, 05C15 (primary), 05C35, 05D40, 05C70 (secondary)
Publisher: 'Cambridge University Press (CUP)'
Year: 2013
DOI identifier: 10.1017/S0963548313000473
OAI identifier: oai:arXiv.org:1205.4171

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