Strong chromatic index of chordless graphs

Abstract

A strong edge colouring of a graph is an assignment of colours to the edges of the graph such that for every colour, the set of edges that are given that colour form an induced matching in the graph. The strong chromatic index of a graph $G$, denoted by $\chi'_s(G)$, is the minimum number of colours needed in any strong edge colouring of $G$. A graph is said to be \emph{chordless} if there is no cycle in the graph that has a chord. Faudree, Gy\'arf\'as, Schelp and Tuza~[The Strong Chromatic Index of Graphs, Ars Combin., 29B (1990), pp.~205--211] considered a particular subclass of chordless graphs, namely the class of graphs in which all the cycle lengths are multiples of four, and asked whether the strong chromatic index of these graphs can be bounded by a linear function of the maximum degree. Chang and Narayanan~[Strong Chromatic Index of 2-degenerate Graphs, J. Graph Theory, 73(2) (2013), pp.~119--126] answered this question in the affirmative by proving that if $G$ is a chordless graph with maximum degree $\Delta$, then $\chi'_s(G) \leq 8\Delta -6$. We improve this result by showing that for every chordless graph $G$ with maximum degree $\Delta$, $\chi'_s(G)\leq 3\Delta$. This bound is tight up to to an additive constant.