Digraphs and variable degeneracy

Abstract: Let $D$ be a digraph, let $p \geq 1$ be an integer, and let $f: V(D) \to \mathbb{N}0^p$ be a vector function with $f=(f_1,f_2,\ldots,f_p)$. We say that $D$ has an $f$-partition if there is a partition $(D_1,D_2,\ldots,D_p)$ into induced subdigraphs of $D$ such that for all $i \in [1,p]$, the digraph $D_i$ is weakly $f_i$-degenerate, that is, in every non-empty subdigraph $D'$ of $D_i$ there is a vertex $v$ such that $\min{d{D'}^+(v), d_{D'}^-(v)} < f_i(v)$. In this paper, we prove that the condition $f_1(v) + f_2(v) + \ldots + f_p(v) \geq \max {d_D^+(v),d_D^-(v)}$ for all $v \in V(D)$ is almost sufficient for the existence of an $f$-partition and give a full characterization of the bad pairs $(D,f)$. Moreover, we describe a polynomial time algorithm that (under the previous conditions) either verifies that $(D,f)$ is a bad pair or finds an $f$-partition. Among other applications, this leads to a generalization of Brooks' Theorem as well as the list-version of Brooks' Theorem for digraphs, where a coloring of digraph is a partition of the digraph into acyclic induced subdigraphs. We furthermore obtain a result bounding the $s$-degenerate chromatic number of a digraph in terms of the maximum of maximum in-degree and maximum out-degree.