82 research outputs found
k-colored kernels
We study -colored kernels in -colored digraphs. An -colored digraph
has -colored kernel if there exists a subset of its vertices such
that
(i) from every vertex there exists an at most -colored
directed path from to a vertex of and
(ii) for every there does not exist an at most -colored
directed path between them.
In this paper, we prove that for every integer there exists a -colored digraph without -colored kernel and if every directed
cycle of an -colored digraph is monochromatic, then it has a -colored
kernel for every positive integer We obtain the following results for some
generalizations of tournaments:
(i) -colored quasi-transitive and 3-quasi-transitive digraphs have a %
-colored kernel for every and respectively (we conjecture
that every -colored -quasi-transitive digraph has a % -colored kernel
for every , and
(ii) -colored locally in-tournament (out-tournament, respectively)
digraphs have a -colored kernel provided that every arc belongs to a
directed cycle and every directed cycle is at most -colored
Tournaments with kernels by monochromatic paths
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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