k-多数竞赛图中支配集的线性下界
A Linear Lower Bound for Dominating Sets in $k$-Majority Tournaments
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中文总结 AI 辅助
研究k-多数竞赛图中最小支配集大小的最大值F(k)，通过证明得到对于k≥3，F(k)有线性下界F(k)≥⌊(k + 1)/2⌋ ，改进了之前Alon等人给出的F(k)的上下界。
中文摘要 AI 辅助
有限顶点集上的k-多数竞赛图由2k - 1个线性序定义，当u在至少k个序中位于v上方时，有u→v。设F(k)是所有k-多数竞赛图中最小支配集大小的最大值。Alon等人证明了对于合适的正的常数C1和C2，有C1k/log k ≤ F(k) ≤ C2k log k。本文证明了对于k≥3，有线性下界F(k)≥⌊(k + 1)/2⌋。
英文摘要
A $k$-majority tournament on a finite vertex set is defined by $2k-1$ linear orders, with $u\to v$ when $u$ lies above $v$ in at least $k$ of the orders. Let $F(k)$ be the maximum, over all $k$-majority tournaments, of the size of a minimum dominating set. Alon, Brightwell, Kierstead, Kostochka, and Winkler proved that $C_1k/\log k \leq F(k) \leq C_2k\log k$ for suitable positive constants $C_1$ and $C_2$. In this paper, we prove the linear lower bound $F(k)\ge \left\lfloor\frac{k+1}{2}\right\rfloor $ for $k\ge 3$.
