Digraphs and Variable Degeneracy

Let $D$ be a digraph, let $p \geq 1$ an integer, and $f: V(D) \to \mathbb{N}_0^p$ vector function with $f=(f_1,f_2,\ldots,f_p)$. We say that has $f$-partition if there is partition $(V_1,V_2,\ldots,V_p)$ of the vertex set such that, for all $i \in [1,p]$, digraph $D_i=D[V_i]$ weakly $f_i$-degenerate, is, in every nonempty subdigraph $D'$ $D_i$ $v$ $\min\{d_{D'}^+(v), d_{D'}^-(v)\} < f_i(v)$. In this paper, we prove condition $f_1(v) + f_2(v) \cdots f_p(v) \max \{d_D^+(v),d_D^-(v)\}$ $v V(D)$ almost sufficient existence give full characterization bad pairs $(D,f)$. Among other applications, leads to generalization Brooks' theorem as well list-version digraphs, where coloring into acyclic induced subdigraphs. furthermore obtain result bounding $s$-degenerate chromatic number terms maximum in-degree out-degree.