(P6, triangle)-free digraphs have bounded dichromatic number

TL;DR

This paper proves that oriented graphs avoiding certain small substructures, specifically (P6, triangle)-free digraphs, have a bounded dichromatic number, advancing understanding of graph colorings in directed graphs.

Contribution

It establishes that (P6, triangle)-free digraphs have bounded dichromatic number, a step towards a broader conjecture on induced subgraphs and graph coloring.

Findings

(P6, triangle)-free digraphs have bounded dichromatic number

Progress towards a conjecture on induced trees and clique restrictions

Provides bounds for specific classes of oriented graphs

Abstract

The dichromatic number of an oriented graph is the minimum size of a partition of its vertices into acyclic induced subdigraphs. We prove that oriented graphs with no induced directed path on six vertices and no triangle have bounded dichromatic number. This is one (small) step towards the general conjecture asserting that for every oriented tree T and every integer k, any oriented graph that does not contain an induced copy of T nor a clique of size k has dichromatic number at most some function of k and T.