Chickens and Dukes

TL;DR

This paper extends the concept of Kings in tournaments to multipartite graphs by introducing Dukes, proving their existence and structural properties in such graphs, and establishing conditions for their presence.

Contribution

It introduces the concept of Dukes in multipartite tournaments and proves their existence and structural properties, expanding the understanding of dominance in directed multipartite graphs.

Findings

Existence of 3-Dukes in all multipartite tournaments

Either a 1-Duke exists or there are three 2-Dukes or four 3-Dukes in any multipartite tournament

Structural results on Dukes in multipartite graphs

Abstract

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.