Injective chromatic index of sparse graphs

Abstract

A k-injective-edge coloring of a graph G is an assignment of colors, i.e. integers in {1,2,…,k}, to the edges of G such that e1 and e3 receive distinct colors for any three consecutive edges e1, e2, e3 of a path or a 3-cycle. The smallest integer k such that G has an injective-edge coloring is called the injective chromatic index of G, denoted by χi′(G). In this paper,we consider the injective-edge coloring of sparse graph G with ΔG=5, and prove that χi′(G)≤10 (resp., 11, 12, 13) if madG<3714 (resp., 3914, 176, 3).