Fractional strong chromatic index of bipartite graphs

Abstract

The strong chromatic index of a graph G, denoted by s′(G), is the minimum possible number of colors in a coloring of the edges of G such that each color class is an induced matching. The corresponding fractional parameter is denoted by sf′(G).For a bipartite graph G we have sf′(G)≤1.5ΔG2. This follows as an easy consequence of earlier results — the fractional variant of Reed’s conjecture and the theorem by Faudree, Gyárfás, Schelp and Tuza from 1990. Both these results are tight so it may seem that the bound 1.5ΔG2 is best possible.We break this “1.5 barrier”. We prove that sf′(G)≤1.4762ΔG2+ΔG1.5 for every bipartite graph G. The main part of the proof is a structural lemma regarding cliques in L(G)2.