论文标题
未配对多个不相交路径的最低半度条件覆盖了挖掘
A minimum semi-degree condition for unpaired many-to-many disjoint path covers in digraphs
论文作者
论文摘要
对于digraph $ d $，令$δ^{0}（d）= \ min \ {δ^{+}（d），δ^{ - }（d）\} $是$ d $的最小半学位。一组$ k $ vertex-disexhight路径，$ \ {p_ {1}，\ dots，p_ {k} \} $，连接一个分离源集$ s = \ s = \ s = \ {1}，\ dot t_ {k} \} $称为$ d $的不成熟的多与Many $ k $ -disjoint指示路径封面（$ k $ -ddpc，for Short for Short），如果每个$ p_ {j} $ joins $ s_ {j} $ s_ {j} $和$ t_ {（j）$ t_（j）} $ for某些$ cout $ n $ n $ not $ not $ not $ not $ c \ \ c。 $ \ bigCup^{k} _ {j = 1} v（p_ {j}）= v（d）$。 在本文中，我们给出了以下结果的新证据，表明每个digraph $ d $都带有$δ^{0}（d）\ geq \ lceil（n+k） / 2 \ rceil $都有一个不成熟的$ k $ -k $ -dddpc，以加入$ s $ s $ s $ s $ s $ t $ t $ t $ s_ {k} \} $和$ t = \ {t_ {1}，\ dots，t_ {k} \} $。此外，我们证明，当$ n \ geq 3k $时，最低限制的界限是最小半学位的界限。
For a digraph $D$, let $δ^{0}(D) = \min \{δ^{+}(D), δ^{-}(D)\}$ be the minimum semi-degree of $D$. A set of $k$ vertex-disjoint paths, $\{P_{1}, \dots, P_{k}\}$, joining a disjoint source set $S = \{s_{1}, \dots, s_{k}\}$ and sink set $T = \{t_{1}, \dots, t_{k}\}$ is called an unpaired many-to-many $k$-disjoint directed path cover ($k$-DDPC for short) of $D$, if each $P_{j}$ joins $s_{j}$ and $t_{σ(j)}$ for some permutation $σ$ on $\{1, \dots , k\}$ and $\bigcup^{k}_{j=1} V(P_{j}) = V(D)$. In this paper, we give a new proof for the following result that every digraph $D$ with $δ^{0}(D) \geq \lceil (n+k) / 2 \rceil$ has an unpaired many-to-many $k$-DDPC joining any disjoint source set $S$ and sink set $T$, where $S = \{s_{1}, \dots, s_{k}\}$ and $T = \{t_{1}, \dots, t_{k}\}$. Moreover, we show that the bound on the minimum semi-degree is best possible when $n \geq 3k$.
