Common Kings of a Chain of Cycles in a Strong Tournament

Logan Post, Zeyu Zheng

Abstract: It is known that every strong tournament has directed cycles of any length, and thereby strong subtournaments of any size. In this note, we prove that they also can share a common vertex which is a king in all of them. This common vertex can be any king in the whole tournament. Further, the Hamiltonian cycles in them can be recursively constructed by inserting an additional vertex to one arc.