A kernel by properly colored paths of an arc-colored digraph D is a set S of vertices of D such that (i) no two vertices of S are connected by a properly colored directed path in D, and (ii) every vertex outside S can reach S by a properly colored directed path in D. In this paper, we conjecture that every arc-colored digraph with all cycles properly colored has such a kernel and verify the conjecture for unicyclic digraphs, semi-complete digraphs and bipartite tournaments, respectively. Moreover, weaker conditions for the latter two classes of digraphs are given.
a b s t r a c tLet t 1 , . . . , t r ∈ [4, 2q] be any r even integers, where q ≥ 2 and r ≥ 1 are two integers.In this note, we show that every bipartite tournament with minimum outdegree at leastThe special case q = 2 of the result verifies the bipartite tournament case of a conjecture proposed by Bermond and Thomassen, stating that every digraph with minimum outdegree at least 2r − 1 contains at least r vertex-disjoint directed cycles.
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