占优势的锦标赛家族

R. Yuster
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引用次数: 0

摘要

对于具有$h$顶点的锦标赛$H$,其典型密度为$h!2^{-\binom{h}{2}}/aut(H)$,即这是在随机锦标赛中$H$的期望密度。如果对于所有足够大的$n$,存在一个$n$ -顶点锦标赛{\em}$G$,使得$G$中${\mathcal F}$的每个元素的密度比其典型密度大一个常数因子,那么一个$h$ -顶点锦标赛家族${\mathcal F}$是的。对于小$h$来说,描述所有优势家族的特征已经是一个挑战。在这里,我们描述了每个$h$的几个大的显性家族。特别地,对于所有足够大的$h$,我们证明了以下内容:(i)对于所有至少有$5\log h$个顶点的锦标赛$H^*$,包含$H^*$作为子图的所有$h$ -顶点锦标赛族是占优的。(ii)最小反馈弧集最大为$\frac{1}{2}\binom{h}{2}-h^{3/2}\sqrt{\ln h}$的所有$h$ -顶点锦标赛族占主导地位。对于较小的$h$,我们在$5$顶点上构建一个$6$(即$50\%$的)锦标赛的优势族,并且在$h=6,7,8,9$上构建一个规模大于$40\%$的优势族。对于所有$h$,我们提供了一个显性家族的明确构造,该构造被推测为获得$h$顶点上比赛的绝对常数分数。另外还提出了一些有趣的开放性问题。
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Dominant tournament families
For a tournament $H$ with $h$ vertices, its typical density is $h!2^{-\binom{h}{2}}/aut(H)$, i.e. this is the expected density of $H$ in a random tournament. A family ${\mathcal F}$ of $h$-vertex tournaments is {\em dominant} if for all sufficiently large $n$, there exists an $n$-vertex tournament $G$ such that the density of each element of ${\mathcal F}$ in $G$ is larger than its typical density by a constant factor. Characterizing all dominant families is challenging already for small $h$. Here we characterize several large dominant families for every $h$. In particular, we prove the following for all $h$ sufficiently large: (i) For all tournaments $H^*$ with at least $5\log h$ vertices, the family of all $h$-vertex tournaments that contain $H^*$ as a subgraph is dominant. (ii) The family of all $h$-vertex tournaments whose minimum feedback arc set size is at most $\frac{1}{2}\binom{h}{2}-h^{3/2}\sqrt{\ln h}$ is dominant. For small $h$, we construct a dominant family of $6$ (i.e. $50\%$ of the) tournaments on $5$ vertices and dominant families of size larger than $40\%$ for $h=6,7,8,9$. For all $h$, we provide an explicit construction of a dominant family which is conjectured to obtain an absolute constant fraction of the tournaments on $h$ vertices. Some additional intriguing open problems are presented.
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