Cycle reversions and dichromatic number in tournaments

TL;DR

The paper demonstrates that any tournament can be made acyclic or decomposed into strongly connected components by reversing a locally finite sequence of cycles, partially confirming a conjecture of Thomassé.

Contribution

It introduces methods to transform arbitrary tournaments into acyclic or strongly connected structures via cycle reversions, advancing understanding of tournament structure.

Findings

Any tournament has finite strong components after cycle reversions.

Any tournament can be covered by two acyclic sets after cycle reversions.

Provides partial proof for Thomassé's conjecture.

Abstract

We show that if $D$ is a tournament of arbitrary size then $D$ has finite strong components after reversing a locally finite sequence of cycles. In turn, we prove that any tournament can be covered by two acyclic sets after reversing a locally finite sequence of cycles. This provides a partial solution to a conjecture of S. Thomass\'e.