Injective edge coloring of graphs

Abstract

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 ?'i(G) is the minimum number of colors permitted in such a coloring. In this paper, exact values of ?'i(G) for several classes of graphs are obtained, upper and lower bounds for ?'i(G) are introduced and it is proven that checking whether ?'i(G) = k is NP-complete.