竞赛图为高度连通的一个新的连通性界
A new connectivity bound for a tournament to be highly linked
AI总结：
本研究针对竞赛图的k-连通性问题，改进了此前的连通性上界，证明最小出度不低于21k-14的⌈12.5k-6⌉-连通竞赛图是k-连通的，同时优化了相关关键引理。
摘要：
有向图$D$被称为$k$-连通的（$k$-linked），若对$D$中任意两个不交顶点集$\{x_1,x_2,\frac{\ldots}{},x_k\text{和}\}\{y_1,y_2,\ldots,y_k\}$，都存在顶点不交的有向路径$P_1,P_2,\ldots,P_k$，使得对每个$i\in[k]$，$P_i$是从$x_i$到$y_i$的有向路径。Pokrovskiy（《组合理论杂志B辑》，2015）证实了Kühn等人（《伦敦数学会会刊》，2014）的一个猜想，证明了每个$452k$-连通的竞赛图都是$k$-连通的。Meng等人（《欧洲组合学杂志》，2021）改进了这一上界，证明了任意$(40k-31)$-连通的竞赛图都是$k$-连通的。本文中，我们通过证明每个最小出度至少为$21k-14$的$\lceil12.5k-6\rceil$-连通竞赛图都是$k$-连通的，得到了更优的上界。此外，我们改进了由Pokrovskiy（《组合理论杂志B辑》，2015）首次提出、后经Meng等人（《欧洲组合学杂志》，2021）强化的一个关键引理。
英文摘要：
A digraph $D$ is $k$-linked if for any pair of two disjoint sets $\{x_{1},x_{2},\ldots,x_{k}\}$ and $\{y_{1},y_{2},\ldots,y_{k}\}$ of vertices in $D$, there exist vertex disjoint dipaths $P_{1},P_{2},\ldots,P_{k}$ such that $P_{i}$ is a dipath from $x_{i}$ to $y_{i}$ for each $i\in[k]$. Pokrovskiy (JCTB, 2015) confirmed a conjecture of Kühn et al. (Proc. Lond. Math. Soc., 2014) by verifying that every $452k$-connected tournament is $k$-linked. Meng et al. (Eur. J. Comb., 2021) improved this upper bound by showing that any $(40k-31)$-connected tournament is $k$-linked. In this paper, we show a better upper bound by proving that every $\lceil 12.5k-6\rceil$-connected tournament with minimum out-degree at least $21k-14$ is $k$-linked. Furthermore, we improve a key lemma that was first introduced by Pokrovskiy (JCTB, 2015) and later enhanced by Meng et al. (Eur. J. Comb., 2021).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
