Complexity and algorithms for injective edge coloring of graphs

An injective k-edge-coloring of a graph G = (V, E) is an assignment & omega; : E & RARR; {1, 2, ... , k} of colors to the edges of G such that any two edges e and f receive distinct colors if there exists an edge g = xy different from e and f such that e is incident on x and f is incident on y. The minimum value of k for which G admits an injective k-edgecoloring is called the injective chromatic index of G and is denoted by & chi;i (G). Given a graph G and a positive integer k, the INJECTIVE EDGE COLORING PROBLEM is to decide whether G admits an injective k-edge-coloring. It is known that INJECTIVE EDGE COLORING PROBLEM is NP-complete for general graphs. In this paper, we strengthen this result by proving that INJECTIVE EDGE COLORING PROBLEM is NP-complete for bipartite graphs by proving that this problem remains NP-complete for perfect elimination bipartite graphs and star-convex bipartite graphs, which are proper subclasses of bipartite graphs. On the positive side, we propose a linear time algorithm for computing the injective chromatic index of chain graphs, which is a proper subclass of both perfect elimination bipartite graphs and starconvex bipartite graphs.