The [formula omitted]-degenerate chromatic number of a digraph

Abstract

The digraph chromatic number of a directed graph D, denoted χA(D), is the minimum positive integer k such that there exists a partition of the vertices of D into k disjoint sets, each of which induces an acyclic subgraph. For any m≥1, a digraph is weaklym-degenerate if each of its induced subgraphs has a vertex of in-degree or out-degree less than m. We introduce a generalization of the digraph chromatic number, namely χm(D), which is the minimum number of sets into which the vertices of a digraph D can be partitioned so that each set induces a weakly m-degenerate subgraph. We show that for all digraphs D without directed 2-cycles, χm(D)≤2Δ(D)4m+1+O(1). Because χ1(D)=χA(D), we obtain as a corollary that χA(D)≤2/5⋅Δ(D)+O(1). We then use this bound to show that χA(D)≤2/3⋅Δ̃(D)+O(1), substantially improving a bound of Harutyunyan and Mohar that states that χA(D)≤(1−e−13)⋅Δ̃(D) for large enough Δ̃(D).