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 $v\notin K$ 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