Injective edge coloring of sparse graphs

A k-edge coloring of a graph G is a mapping phi : E(G) ->{1, 2,& mldr;,k}. A k-injective edge coloring is a k-edge coloring such that any two edges e and f with distance exactly 1 or in a triangle receive distinct colors. The injective edge chromatic index of a graph G, denoted by chi(inj)'(G), is the minimum integer k such that G has a k-injective edge coloring. In this paper, we show that chi(inj)'(G) <= max{2 Delta(G) - 2, 6} if mad(G) < (5)/ (2); chi(inj)'(G) <= 2 Delta(G) if mad(G) <( 29)/ (11); and chi(inj)'(G) <= 2 Delta(G) + 1 if mad(G) < (14) /(5). We also show that chi(inj)'(G) <= 4 if G is subcubic and mad(G) < (7)/ (3).