3-锦标赛中的统治
Domination in 3-tournaments
摘要：
一个3-竞赛图是一个完全的3-一致超图，其中每条边都有一个特殊的顶点作为它的尾。顶点集$X$支配$T$，如果不在$X$中的每个顶点都包含在其尾部在$X$中的边中。$T$的支配数是这样的$X$的最小尺寸。Gyárfás推广了关于通常（图）竞赛图的著名结果，证明了存在具有任意大控制数的3-竞赛图，并且如果任意四个顶点诱导两个具有相同尾的三元组，则情况并非如此。在这个简短的说明中，我们解决了这两个问题，证明了第一个猜想，并反驳了第二个。
英文摘要：
A 3-tournament is a complete 3-uniform hypergraph where each edge has a special vertex designated as its tail. A vertex set $X$ dominates $T$ if every vertex not in $X$ is contained in an edge whose tail is in $X$. The domination number of $T$ is the minimum size of such an $X$. Generalizing well-known results about usual (graph) tournaments, Gyárfás conjectured that there are 3-tournaments with arbitrarily large domination number, and that this is not the case if any four vertices induce two triples with the same tail. In this short note we solve both problems, proving the first conjecture and refuting the second.
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。
