82 research outputs found

    k-colored kernels

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    We study kk-colored kernels in mm-colored digraphs. An mm-colored digraph DD has kk-colored kernel if there exists a subset KK of its vertices such that (i) from every vertex v∉Kv\notin K there exists an at most kk-colored directed path from vv to a vertex of KK and (ii) for every u,v∈Ku,v\in K there does not exist an at most kk-colored directed path between them. In this paper, we prove that for every integer k≥2k\geq 2 there exists a (k+1)% (k+1)-colored digraph DD without kk-colored kernel and if every directed cycle of an mm-colored digraph is monochromatic, then it has a kk-colored kernel for every positive integer k.k. We obtain the following results for some generalizations of tournaments: (i) mm-colored quasi-transitive and 3-quasi-transitive digraphs have a kk% -colored kernel for every k≥3k\geq 3 and k≥4,k\geq 4, respectively (we conjecture that every mm-colored ll-quasi-transitive digraph has a kk% -colored kernel for every k≥l+1)k\geq l+1), and (ii) mm-colored locally in-tournament (out-tournament, respectively) digraphs have a kk-colored kernel provided that every arc belongs to a directed cycle and every directed cycle is at most kk-colored

    Tournaments with kernels by monochromatic paths

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    In this paper we prove the existence of kernels by monochromatic paths in m-coloured tournaments in which every cyclic tournament of order 3 is atmost 2-coloured in addition to other restrictions on the colouring ofcertain subdigraphs. We point out that in all previous results on kernelsby monochromatic paths in arc coloured tournaments, certain smallsubstructures are required to be monochromatic or monochromatic with atmost one exception, whereas here we allow up to three colours in two smallsubstructures
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