Following on the King Chicken Theorems originally proved by Maurer, we examine the idea of multiple flocks of chickens by bringing the chickens from tournaments to multipartite tournaments. As Kings have already been studied in multipartite settings, notably by Koh-Tan and Petrovic-Thomassen, we examine a new type of chicken more suited than Kings for these multipartite graphs: Dukes. We define an M-Duke to be a vertex from which any vertex in a different partite set is accessible by a directed path of length at most M. In analogy with Maurer's paper, we prove various structural results regarding Dukes. In particular, we prove the existence of 3-Dukes in all multipartite tournaments, and we conclude by proving that in any multipartite tournament, either there is a 1-Duke, three 2-Dukes, or four 3-Dukes.
翻译:在莫伊雷尔最初证明的 " 鸡王神话 " 上,我们研究了多只鸡群的想法,把从锦标赛到多方锦标赛的鸡群带到多方锦标赛。正如国王们已经在多个方面,特别是Koh-Tan和Petrovic-Thomassen的场合中研究过一样,我们检查了比国王们更适合这些多方图的新型鸡:杜克斯。我们把M-Duke定义为一个脊椎,不同部分的任何脊椎都可以通过最长的M.的直线路径进入。比喻莫伊雷尔的论文,我们证明了关于公爵的各种结构性结果。特别是,我们证明在所有多方锦标赛中都存在3Dukes,我们的结论是,在任何多方的锦标赛中,要么是1-Duke,32-Dukes,要么是4-3Dukes。