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