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- W4214845824 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 $(V_1,V_2,ldots,V_p)$ of the vertex set of $D$ such that, for all $i in [1,p]$, the digraph $D_i=D[V_i]$ is weakly $f_i$-degenerate, that is, in every nonempty 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 $f_1(v) + f_2(v) + cdots + f_p(v) geq max {d_D^+(v),d_D^-(v)}$ 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 $(D,f)$. 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 a 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." @default.
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- W4214845824 date "2022-03-01" @default.
- W4214845824 modified "2023-10-18" @default.
- W4214845824 title "Digraphs and Variable Degeneracy" @default.
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- W4214845824 doi "https://doi.org/10.1137/20m1386827" @default.
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