arXiv 1202.40172012-02-17math.CO

半完全多部有向图的$k$着色核

$k$-colored kernels in semicomplete multipartite digraphs

Hortensia Galeana-Sánchez, Bernardo Llano, Juan José Montellano-Ballesteros

摘要：

一个$m$着色有向图$D$有$k$着色核，如果存在其顶点的子集$K $，使得对于每个顶点$v\notin K$，存在从$v$到$K$的顶点的至多$k$着色有向路，并且对于每个$% u,v\in K$，它们之间不存在至多$k$着色有向路.本文证明了一个$m$着色的半完全$r$ -部有向图$D$有一个$k$着色的核，条件是$r\geq 3$和{enumerate} [（i）] $k\geq 4,$ [（ii）] $k=3$，且包含在$D$中的每个$\overrightarrow{C}_{4}$至多是2-色的，且包含在$D$中的每个$\overrightarrow{C}_{5}$至多是3-色的或包含在$D$中的每个$\overrightarrow{C}_{3}\uparrow \overrightarrow{C}_{3}$至多是2-色的.[（iii）] $k=2$，并且$D$中包含的每个$\overrightarrow{C}_{3}$和$\overrightarrow{C}%_{4}$都是单色的。{enumerate}若$D$是$m$着色的半完全二部图，且$k=2$（分别为. $k=3 $），并且$D$中所含的每个$\overrightarrow{C}_{4}\upuparrows \overrightarrow{C}_{4}$至多为2种颜色（分别为3色），则$D$具有$% 2$色（分别为3色）内核。利用这些结果和以前的结果，我们得到了$m$ -着色半完全$r$ -部有向图中存在$k$ -着色核的条件，对于每个$k\geq 2$和$r\geq 2$。

英文摘要：

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 $r\geq 2$.

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摘要：

英文摘要：

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