Injective edge-coloring of graphs with given maximum degree

Abstract

A coloring of edges of a graph G is injective if for any two distinct edges e1 and e2, the colors of e1 and e2 are distinct if they are at distance 1 in G or in a common triangle. Naturally, the injective chromatic index of G, χinj′(G), is the minimum number of colors needed for an injective edge-coloring of G. We study how large can be the injective chromatic index of G in terms of maximum degree of G when we have restrictions on girth and/or chromatic number of G. We also compare our bounds with analogous bounds on the strong chromatic index.