$k$-colored kernels in semicomplete multipartite digraphs

TL;DR

This paper establishes conditions under which $k$-colored kernels exist in semicomplete multipartite digraphs, extending previous results and providing new criteria based on cycle colorings and graph structure.

Contribution

The paper introduces new sufficient conditions for the existence of $k$-colored kernels in semicomplete multipartite digraphs, generalizing prior work and covering various cases based on cycle colorings.

Findings

Proves existence of $k$-colored kernels for $r extgreater=3$ under specific cycle coloring conditions.

Establishes $2$-colored and $3$-colored kernel existence in semicomplete bipartite digraphs with cycle coloring constraints.

Provides comprehensive conditions for $k$-colored kernels in $m$-colored semicomplete $r$-partite digraphs.

Abstract

An $m$-colored digraph $D$ has $k$-colored kernel if there exists a subset $K $ of its vertices such that for every vertex $v\notin K$ there exists an at most $k$-colored directed path from $v$ to a vertex of $K$ and for every $% u,v\in K$ there does not exist an at most $k$-colored directed path between them. In this paper we prove that an $m$-colored semicomplete $r$-partite digraph $D$ has a $k$-colored kernel provided that $r\geq 3$ and {enumerate} [(i)] $k\geq 4,$ [(ii)] $k=3$ and every $\overrightarrow{C}_{4}$ contained in $D$ is at most 2-colored and, either every $\overrightarrow{C}_{5}$ contained in $D$ is at most 3-colored or every $\overrightarrow{C}_{3}\uparrow \overrightarrow{C}_{3}$ contained in $D$ is at most 2-colored, [(iii)] $k=2$ and every $\overrightarrow{C}_{3}$ and $\overrightarrow{C}%_{4}$ contained in $D$ is monochromatic. {enumerate} If $D$ is an $m$-colored semicomplete bipartite digraph and $k=2$ (resp. $k=3 $) and every $\overrightarrow{C}_{4}\upuparrows \overrightarrow{C}_{4}$ contained in $D$ is at most 2-colored (resp. 3-colored), then $D$ has a $% 2$-colored (resp. 3-colored) kernel. Using these and previous results, we obtain conditions for the existence of $k$-colored kernels in $m$-colored semicomplete $r$-partite digraphs for every $k\geq 2$ and $r\geq 2$.