The distance-t chromatic index of graphs

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 $(2-\text{eps}){\mathrm{\Delta }}^t$ for graphs of maximum degree at most $\mathrm{\Delta }$, where $\text{eps}$ is some absolute positive constant independent of $t$. The other is a bound of $O(\Delta^t/\log \Delta)$(as$\Delta\to\infty$)forgraphsofmaximumdegreeatmost$\Delta$andgirthatleast$2t+1$. 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 $\mathrm{\Delta }$, with distance-$t$ chromatic index at least $\mathrm{\Omega }({\mathrm{\Delta }}^t\mathrm{/}\log \mathrm{\Delta })$.