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- W2950333782 abstract "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 $vnotin K$ there exists an at most $k$-colored directed path from $v$ to a vertex of $K$ and (ii) for every $u,vin K$ there does not exist an at most $k$-colored directed path between them. In this paper, we prove that for every integer $kgeq 2$ 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 $kgeq 3$ and $kgeq 4,$ respectively (we conjecture that every $m$-colored $l$-quasi-transitive digraph has a $k$% -colored kernel for every $kgeq l+1)$, 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." @default.
- W2950333782 created "2019-06-27" @default.
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- W2950333782 date "2012-01-13" @default.
- W2950333782 modified "2023-09-29" @default.
- W2950333782 title "k-colored kernels" @default.
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