From Edge-Coloring to Strong Edge-Coloring

In this paper we study a generalization of both proper edge-coloring and strong edge-coloring: k-intersection edge-coloring, introduced by Muthu, Narayanan and Subramanian [18]. In this coloring, the set S(v) of colors used by edges incident to a vertex v does not intersect S(u) on more than k colors when u and v are adjacent. We provide some sharp upper and lower bounds for χk-int for several classes of graphs. For l-degenerate graphs we prove that χk-int(G) 6 (l + 1)∆− l(k − 1)− 1. We improve this bound for subcubic graphs by showing that χ2-int(G) 6 6. We show that calculating χk-int(Kn) for arbitrary values of k and n is related to some problems in combinatorial set theory and we provide bounds that are tight for infinitely many values of n. Furthermore, for complete bipartite graphs we prove that χk-int(Kn,m) = ⌈ mn k ⌉ . Finally, we show that computing χk-int(G) is NP-complete for every k > 1.