Injective edge coloring of sparse graphs with maximum degree 5

A k-injective-edge coloring of a graph G is a mapping c : E(G)-* {1 , 2 , middot middot middot, k} such that c(e(1)) &NOTEQUexpressionL;c(e(3)) for any three consecutive edges e1 , e2 , e3 of a path or a 3-cycle. chi i(G) = min{k|G has a k-injective-edge coloring} is called the injective chromatic index of G. In this paper, we prove that for graphs G with delta(G) < 5, (1) chi i(G) < 8 if mad(G) < 7/3; (2) chi i(G) < 9 if mad(G) < 12/5; (3) chi i(G) < 10 if mad(G) < 5/2 ; (4) chi i(G) < 11 if mad(G) < 18/7.