On coloring digraphs with forbidden induced subgraphs

TL;DR

This paper proves a conjecture about coloring certain oriented graphs with forbidden subgraphs, showing they can be partitioned into two acyclic parts, and extends this to broader classes, advancing understanding of directed graph colorings.

Contribution

It proves a key conjecture for a class of oriented graphs and extends the result to larger classes defined by forbidden subgraphs, advancing graph coloring theory.

Findings

Oriented graphs with transitive out-neighborhoods can be partitioned into two acyclic subgraphs.

Extensions to larger classes of digraphs with forbidden induced subgraphs are established.

Several special cases of a directed extension of the Gyárfás-Sumner conjecture are resolved.

Abstract

We prove a conjecture by Aboulker, Charbit and Naserasr by showing that every oriented graph in which the out-neighborhood of every vertex induces a transitive tournament can be partitioned into two acyclic induced subdigraphs. We prove multiple extensions of this result to larger classes of digraphs defined by a finite list of forbidden induced subdigraphs. We thereby resolve several special cases of an extension of the famous Gy\'{a}rf\'{a}s-Sumner conjecture to directed graphs by Aboulker et al.