Strong chromatic index of graphs

A strong edge coloring of a graph G is a coloring of edges of G such that each color class is an induced matching in G, and the strong chromatic index of G, denoted s′(G), is the minimum number of colors in a strong edge coloring of G. We also consider the fractional and topological variant of strong chromatic index, denoted sf (G) and s ′ t(G) respectively. Our dream goal is to give a sharp upper bound on the strong chromatic index of a graph G with the given maximum degree ∆. A simple, greedy argument shows that s′(G) ≤ 2∆, and the best known bound is 1.93∆ (Bruhn and Joos, 2015+). This result is still far from 5 4 ∆, conjectured by Erd®s and Ne2et°il in 1985 (which would be sharp). For bipartite graphs the conjectured bound is s′(G) ≤ ∆ (Faudree, Gyárfás, Schelp and Tuza, 1989) and the best known is s′(G) ≤ 1.93∆ (that is, there is no improvement over the mentioned result of Bruhn and Joos); it follows that st(G) ≤ 1.93∆. For fractional strong chromatic index, a better bound 1.5∆ can be obtained from earlier results. Our main contribution is breaking the 1.5∆ boundary we show that for a bipartite graph G of maximum degree ∆ we have sf (G) ≤ 1.476∆. Moreover, we signi cantly improve the bound on the topological variant: for a bipartite graph G of maximum degree ∆ we show st(G) ≤ 1.703∆. We also show that if G is a graph such that every edge of G is in at most ∆ 2 f 4-cycles, then s′(G) ≤ K ∆2 ln f for some absolute constant K, and give a bound s′(G) ≤ 4∆− 3 in case when G is chordless.