  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 $v\notin K$ there exists an at most $k$-colored
directed path from $v$ to a vertex of $K$ and
  (ii) for every $u,v\in K$ there does not exist an at most $k$-colored
directed path between them.
  In this paper, we prove that for every integer $k\geq 2$ 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 $k\geq 3$ and $k\geq 4,$ respectively (we conjecture
that every $m$-colored $l$-quasi-transitive digraph has a $k$% -colored kernel
for every $k\geq l+1)$, 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.
