K-colored Kernels

We study k-colored kernels in m-colored digraphs. An m-colored digraph D has k-colored kernel if there exists a subset K of its vertices such that (i) from every vertex v / ∈ K there exists an at most k-colored directed path from v to a vertex of K and (ii) for every u, v ∈ K there does not exist an at most k-colored directed path between them. In this paper, we prove that for every integer k ≥ 2 there exists a (k + 1)-colored digraph D without k-colored kernel and if every directed cycle of an m-colored digraph is monochromatic, then it has a k-colored kernel for every positive integer k. We obtain the following results for some generalizations of tournaments: (i) m-colored quasi-transitive and 3-quasi-transitive digraphs have a k-colored kernel for every k ≥ 3 and k ≥ 4, respectively (we conjecture that every m-colored l-quasi-transitive digraph has a k-colored kernel for every k ≥ l + 1), and (ii) m-colored locally in-tournament (out-tournament, respectively) digraphs have a k-colored kernel provided that every arc belongs to a directed cycle and every directed cycle is at most k-colored.