On coloring digraphs with forbidden induced subgraphs

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 Gyárfás-Sumner conjecture to

Proving a Directed Analogue of the Gyárfás-Sumner Conjecture for Orientations of $P_4$

“…In this paper, we show that every orientation of P 4 is − → χ -bounding and thus the ACN − → χ -boundedness conjecture holds for all orientations of Our result also answers the question of [25] in the affirmative, that is, for H ∈ { − → P 4 , − → A 4 } and any k 4 the class of H-free oriented graphs not containing a transitive tournament of order k has bounded dichromatic number.…”

“…− → K t denote the transitive tournament on t vertices. In [25], Steiner showed that the class of ( − → K 3 , − → A 4 )-free oriented graphs has bounded dichromatic number. In the same paper Steiner asked whether the class of (H, − → K t )-free oriented graphs has bounded dichromatic number for t 4 and…”

Heroes in oriented complete multipartite graphs

“…In [14] it is proved that it is enough to prove the conjecture for trees, the conjecture has been proved to be true for oriented stars [8].…”

“…As said in Section 5.1, it is proved in [8] that for every oriented star F , all transitive tournaments are heroes in Forb F ( ) ind . The only other known result so far is concerned with ⎯→ ⎯ K 1,2 (the oriented star on three vertices, with one vertex of out-degree 2 and two vertices of in-degree 1): it is proved in [1,14]…”

“…As said in Section 5.1, it is proved in [8] that for every oriented star $$F$$, all transitive tournaments are heroes in $$For{b}_{\mathrm{ind}}(F)$$. The only other known result so far is concerned with $${\overrightarrow{K}}_{1,2}$$ (the oriented star on three vertices, with one vertex of out‐degree 2 and two vertices of in‐degree 1): it is proved in [1, 14] that $${K}_{1}\Rightarrow {C}_{3}$$ (and thus $${C}_{3}$$ too) is a hero in $$For{b}_{\mathrm{ind}}({\overrightarrow{K}}_{1,2})$$. Note that $${\overrightarrow{P}}_{3}$$ is an oriented star.…”