关于Sauer-Spender定理的有向形式
On directed version of the Sauer-Spender Theorem
摘要：
设$D=(V,A)$是$n$阶的有向图，$W$是$V$的任意子集.定义$D$中$W$的最小半度为$δ^0(W)=\mbox{min}\{δ^+(W),δ^-(W)\}$，其中$δ^+(W)$为$D$中$W$的最小出度，$δ^-(W)$为$D$中$W$的最小入度.令$k$是具有$k\geq 1$的整数。本文证明了：对任何正整数划分$|W|=\sum_{i=1}^{k}n_i$，每个$i$都有$n_i\geq 2$，若$δ^0(W)\geq \frac{3n-3}{4}$，则在$D$中存在$k$顶点不交圈$C_1,\ldots,C_k$，使得每个$C_i$恰包含$W$的$n_i$顶点.此外，如果$k=1$，则$δ^0(W)$的下限可以改进为$\frac{n}{2}$，如果$n\geq 2|W|$，则可以改进为$\frac{n}{2}+|W|-1$。最小半度条件$δ^0(W)\geq \frac{3n-3}{4}$在某种意义上是尖锐的，这一结果部分证实了Wang [Graphs and Combinatorics 16（2000）453-462]提出的猜想.它也是一个有向版本的Sauer-Spender定理的顶点不相交圈图[J. Combin.理论B，25（1978）295-302]。
英文摘要：
Let $D=(V,A)$ be a digraph of order $n$ and let $W$ be any subset of $V$. We define the minimum semi-degree of $W$ in $D$ to be $δ^0(W)=\mbox{min}\{δ^+(W),δ^-(W)\}$, where $δ^+(W)$ is the minimum out-degree of $W$ in $D$ and $δ^-(W)$ is the minimum in-degree of $W$ in $D$. Let $k$ be an integer with $k\geq 1$. In this paper, we prove that for any positive integer partition $|W|=\sum_{i=1}^{k}n_i$ with $n_i\geq 2$ for each $i$, if $δ^0(W)\geq \frac{3n-3}{4}$, then there are $k$ vertex disjoint cycles $C_1,\ldots,C_k$ in $D$ such that each $C_i$ contains exactly $n_i$ vertices of $W$. Moreover, the lower bound of $δ^0(W)$ can be improved to $\frac{n}{2}$ if $k=1$, and $\frac{n}{2}+|W|-1$ if $n\geq 2|W|$. The minimum semi-degree condition $δ^0(W)\geq \frac{3n-3}{4}$ is sharp in some sense and this result partially confirms the conjecture posed by Wang [Graphs and Combinatorics 16 (2000) 453-462]. It is also a directed version of the Sauer-Spender Theorem on vertex disjoint cycles in graphs [J. Combin. Theory B, 25 (1978) 295-302].
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