Extension of Gyarfas-Sumner conjecture to digraphs

Abstract

The dichromatic number of a digraph D is the minimum number of colors needed to color its vertices in such a way that each color class induces an acyclic digraph. As it generalizes the notion of the chromatic number of graphs, it has been a recent center of study. In this work we look at possible extensions of Gyárfás-Sumner conjecture. More precisely, we propose as a conjecture a simple characterization of finite sets F of digraphs such that every oriented graph with sufficiently large dichromatic number must contain a member of F as an induce subdigraph.Among notable results, we prove that oriented triangle-free graphs without a directed path of length 3 are 2-colorable. If condition of "triangle-free" is replaced with "K 4 -free", then we have an upper bound of 414. We also show that an orientation of complete multipartite graph with no directed triangle is 2-colorable. To prove these results we introduce the notion of nice sets that might be of independent interest.

On coloring digraphs with forbidden induced subgraphs

“…Just as in the undirected case, Aboulker et al [1] observed several necessary conditions for a finite set F of digraphs to be heroic, which we summarize in the following. Proposition 1.3 (cf.…”

“…In the same spirit, Aboulker, Charbit and Naserasr [1] recently initiated the systematic study of the relation between excluded induced subdigraphs and the dichromatic number and asked the following intriguing question. Problem 1.2.…”

“…Under which circumstances does there exist C ∈ N such that every digraph D without an induced subdigraph isomorphic to a member of F satisfies χ(D) ≤ C? Following the terminology introduced by Aboulker et al [1], we denote by Forb ind (F) the set of digraphs containing no induced subdigraph isomorphic to a member of F. We say that F is heroic if the digraphs in Forb ind (F) have bounded dichromatic number and in this case we denote χ(Forb ind (F)) := max{ χ(D)|D ∈ Forb ind (F)}.…”

“…In contrast to undirected graphs, only heroic sets of size at least 3 are interesting to consider, as the necessary conditions from Proposition 1.3 directly imply that { ↔ K 2 , K 2 } is the only heroic set of size two (and is so trivially). Aboulker et al in [1] proved that every heroic set of size three must be of one of the following types:…”

“…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].…”