Strong Chromatic Index of Chordless Graphs

A strong edge coloring of a graph is an assignment of colors to the edges of the graph such that for every color, the set of edges that are given that color 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 colors needed in any strong edge coloring of G. A graph is said to be chordless if there is no cycle in the graph that has a chord. Faudree, Gyarfas, Schelp, and Tuza (The Strong Chromatic Index of Graphs, Ars Combin 29B (1990), 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), 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) <= 8 Delta-6. We improve this result by showing that for every chordless graph G with maximum degree Delta, chi(s)'(G) <= 3 Delta. This bound is tight up to an additive constant.