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− ε)∆t for graphs of maximum degree at most ∆, where ε is some absolute positive constant independent of t. The other is a bound of O(∆t/ log ∆) (as ∆→∞) for graphs of maximum degree at most ∆ 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 ∆, with distance-t chromatic index at least Ω(∆t/ log ∆). ∗Supported by project P202/12/G061 of the Czech Science Foundation. †Supported by a NWO Veni Grant. Part of this work was carried out while this author was at Durham University, supported by EPSRC, grant EP/G066604/1. 1 ar X iv :1 20 5. 41 71 v3 [ m at h. C O ] 3 S ep 2 01 3