论文标题
$ H $ -KERNELS中的$ H $ -KERNELS $ H $ -CORORED DIGRAPHS没有$（ξ_{1}，ξ，ξ__{2}）$ - $ h $ -h $ -subdivision
$H$-kernels in $H$-colored digraphs without $(ξ_{1}, ξ, ξ_{2})$-$H$-subdivisions of $\overrightarrow{C_{3}}$
论文作者
论文摘要
令$ h $为循环的挖掘物，而无循环的$ d $ d $ d $ c：a（d）\ rightArrow v（h）$（$ d $ a（$ d $）据说是$ h $颜色的digraph）。 $ d $中的定向路径$ w $被据说是$ h $ path的，并且仅当$ w $上遇到的连续颜色在$ h $中遇到的指示步行。如果（1）在$ n $中的每对不同的顶点（1），它们之间没有$ h $ - n $ h $ - n $ h $ path在v（$ d $ d $ d $ \ setminus $ n $ in $ h $ h $ d $ d $ d $ d $ d $ d $ n $ d $ n $ n $ d $ n $ n $ d $ d $ d $ n $ n $ n $ d $ n $ d $ d $ d $ n $ n $ d $ n $ d $ d $ d $，在此定义下，每当$ a（h）= \ emptySet $时，$ h $ -KERNEL是内核。 $ d $ $ d $的颜色级digraph $ \ mathscr {c} _c $（$ d $）是digraph，其顶点是$ d $的颜色和（$ i $，$ j $，$ j $）$ （$ v $，$ w $）$ d $，因此（$ u $，$ v $）具有颜色$ i $，而（$ v $，$ w $）具有颜色$ j $。由于并非每个$ h $颜色的digraph都具有$ h $ -KERNEL和$ v（\ MATHSCR {C} _C（d））= V（H）$，因此自然的问题是：$ \ Mathscr {C} _c（d）$的结构性属性 在本文中，我们通过$ v（h）$的分区$ξ$和分区\ {$ξ_1$，$ξ_2$ \}来研究$ h $ -KERNEL的问题。相对于分区\ {$ξ_1$，$ξ_2$ \}，我们在有向周期和Digraph $ d $的定向路径上建立了条件。特别是，我们注意分区$ξ$和\ {$ξ_1$，$ξ_2$ \}产生的某些亚热，即$（ξ_{1}，ξ，ξ__{2}）$ - $ - $ - $ - $ h $ - ξ_{2}）$ - $ h $ -subdivisions的$ \ oferrightArrow {p_ {3}} $。 我们举了一些例子，表明主要结果中的每个假设都很紧。
Let $H$ be a digraph possibly with loops and $D$ a digraph without loops with a coloring of its arcs $c:A(D) \rightarrow V(H)$ ($D$ is said to be an $H$-colored digraph). A directed path $W$ in $D$ is said to be an $H$-path if and only if the consecutive colors encountered on $W$ form a directed walk in $H$. A subset $N$ of vertices of $D$ is said to be an $H$-kernel if (1) for every pair of different vertices in $N$ there is no $H$-path between them and (2) for every vertex $u$ in V($D$)$\setminus$$N$ there exists an $H$-path in $D$ from $u$ to $N$. Under this definition an $H$-kernel is a kernel whenever $A(H)=\emptyset$. The color-class digraph $\mathscr{C}_C$($D$) of $D$ is the digraph whose vertices are the colors represented in the arcs of $D$ and ($i$,$j$) $\in$ $A$($\mathscr{C}_C$($D$)) if and only if there exist two arcs, namely ($u$,$v$) and ($v$,$w$) in $D$, such that ($u$,$v$) has color $i$ and ($v$,$w$) has color $j$. Since not every $H$-colored digraph has an $H$-kernel and $V(\mathscr{C}_C(D))= V(H)$, the natural question is: what structural properties of $\mathscr{C}_C(D)$, with respect to the $H$-coloring, imply that $D$ has an $H$-kernel? In this paper we investigate the problem of the existence of an $H$-kernel by means of a partition $ξ$ of $V(H)$ and a partition \{$ξ_1$, $ξ_2$\} of $ξ$. We establish conditions on the directed cycles and the directed paths of the digraph $D$, with respect to the partition \{$ξ_1$, $ξ_2$\}. In particular we pay attention to some subestructures produced by the partitions $ξ$ and \{$ξ_1$, $ξ_2$\}, namely $(ξ_{1}, ξ, ξ_{2})$-$H$-subdivisions of $\overrightarrow{C_{3}}$ and $(ξ_{1}, ξ, ξ_{2})$-$H$-subdivisions of $\overrightarrow{P_{3}}$. We give some examples which show that each hypothesis in the main result is tight.
