k-colored kernels
Hortensia Galeana-S\'anchez, Bernardo Llano, Juan Jos\'e, Montellano-Ballesteros

TL;DR
This paper investigates the existence of k-colored kernels in m-colored digraphs, establishing conditions under which such kernels exist or do not, and exploring specific classes like quasi-transitive digraphs.
Contribution
It proves the existence and non-existence results of k-colored kernels in various m-colored digraph classes, extending kernel theory in colored digraphs.
Findings
Existence of (k+1)-colored digraphs without k-colored kernels for k≥2.
Monochromatic cycles guarantee k-colored kernels for all k.
Certain classes like quasi-transitive digraphs have k-colored kernels for all k beyond a threshold.
Abstract
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…
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Taxonomy
TopicsAdvanced Graph Theory Research · Limits and Structures in Graph Theory · Game Theory and Applications
