Injective edge chromatic number of sparse graphs

Abstract

A k-edge coloring ϕ of graph G is injective if any two edges at distance 2 or in the same triangle get different colors. The minimum k in such an edge coloring is the injective edge chromatic number of G, written as χi′(G). We prove in this paper that for any graph G with Δ(G)≤5, χi′(G)≤18 if mad(G)<165, χi′(G)≤19 if mad(G)<72 and χi′(G)≤20 if mad(G)<154.