Hamiltonicity in infinite tournaments

TL;DR

This paper proves that all countable tournaments have a topological Hamilton path, and strongly connected ones have a topological Hamilton circle, extending finite tournament theorems to the infinite case in a topological setting.

Contribution

It establishes the existence of topological Hamilton paths and circles in the compactification of infinite tournaments, a novel extension of finite tournament results.

Findings

Countable tournaments have a topological Hamilton path in their compactification.

Strongly connected tournaments have a topological Hamilton circle.

Finite tournament theorems do not directly extend to infinite tournaments without topological considerations.

Abstract

We prove that for all countable tournaments $D$ the recently discovered compactification $|D|$ by their ends and limit edges contains a topological Hamilton path: a topological arc that contains every vertex. If $D$ is strongly connected, then $|D|$ contains a topological Hamilton circle. These results extend well-known theorems about finite tournaments, which we show do not extend to the infinite in a purely combinatorial setting.