Common kings of a chain of cycles in a strong tournament

TL;DR

This paper proves that in a strong tournament, there exists a common vertex that is a king in all cycles of a chain, and shows how Hamiltonian cycles can be constructed recursively by inserting vertices.

Contribution

It establishes the existence of a universal king vertex shared by all cycles in a strong tournament and introduces a recursive method for constructing Hamiltonian cycles.

Findings

Existence of a common king vertex in all cycles of a strong tournament

Recursive construction of Hamiltonian cycles by inserting vertices

Universal king vertex can be any king in the entire tournament

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 of 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 directed edge.