Reducing the dichromatic number via cycle reversions in infinite digraphs

TL;DR

This paper proves that for any (possibly infinite) directed graph, it is possible to reorient edges through cycle reversions to ensure the resulting graph has a dichromatic number of at most two, with finite local edge-connectivities in each strong component.

Contribution

It confirms Thomassé's conjecture by showing a finite-cycle reversion process can reduce the dichromatic number in infinite digraphs while maintaining finite local edge-connectivities.

Findings

Reorientation process guarantees dichromatic number ≤ 2.

Each edge is reversed only finitely many times.

Finite local edge-connectivities are preserved in strong components.

Abstract

We prove the following conjecture of S. Thomass\'e: for every (potentially infinite) digraph $ D $ it is possible to iteratively reverse directed cycles in such a way that the dichromatic number of the final reorientation $ D^{*} $ of $ D $ is at most two and each edge is flipped only finitely many times. In addition, we guarantee that in every strong component of $ D^{*} $ all the local edge-connectivities are finite and any edge is reversed at most twice.