Injective edge coloring of sparse graphs

A k-injective edge coloring of a graph G is a coloring f : E(G) -> C = {1, 2, 3,..., k}, such that if e(1), e(2) and e(3) are consecutive edges in G, then f(e(1)) not equal f(e(3)). chi(i)' (G) = min{k vertical bar G has a k-injective edge coloring} is called the injective edge coloring number. In this paper, we consider the upper bound of chi(i)' (G) in terms of the maximum average degree mad(G), where mad(G) = max {2 vertical bar E(H)vertical bar / vertical bar V(H)vertical bar , H subset of G}.