Injective edge coloring of graphs

Three edges e₁, e₂ and e₃ in a graph G are consecutive if they form a path (in this order) or a cycle of lengths three. An injective edge coloring of a graph G = (V, E) is a coloring c of the edges of G such that if e₁, e₂ and e₃ are consecutive edges in G, then c(e₁) ≠ c(e₃). The injective edge coloring number χ^′ _i ^(G) is the minimum number of colors permitted in such a coloring. In this paper, exact values of χ^′ _i(G) for several classes of graphs are obtained, upper and lower bounds for χ^′ _i (G) are introduced and it is proven that checking whether χ^′ _i (G) = k is NP-complete.