On coloring digraphs with forbidden induced subgraphs

Abstract

We prove a conjecture by Aboulker, Charbit and Naserasr [1] 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 directed graphs stated in [1].

Heroes in orientations of chordal graphs

“…It is proved in [9] that heroes in F orb ind (K k ) are the same as heroes in tournaments, where K k is the graph on k vertices with no arc (which is in particular the simplest union of disjoint oriented stars). In [1] and [14], it is proved that K 1 ⇒ C 3 is a hero in F orb ind ( K 1,2 ), where K 1,2 is the star on three vertices with a vertex of out-degree 2. In [2], heroes in the class of orientations of complete multipartite graphs (which corresponds to the class F orb ind (K 1 + K 2 ) where K 1 + K 2 is the graph made of an isolated vertex and an arc) are almost fully characterized, up to one particular digraph, namely ∆(1, 2, 2).…”