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- W4291335965 abstract "The digraph chromatic number of a directed graph $D$, denoted $chi_A(D)$, 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 $m geq 1$, a digraph is weakly $m$-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 $chi_m(D)$, 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, $chi_m(D) leq frac{2{Delta}(D)}{4m+1} + O(1)$. Because $chi_1(D) = chi_A(D)$, we obtain as a corollary that $chi_A(D) leq 2/5 cdot {Delta}(D) + O(1)$. We then use this bound to show that $chi_A(D) leq sqrt{2/3} cdot tilde{Delta}(D) + O(1)$, substantially improving a bound of Harutyunyan and Mohar that states that $chi_A(D) leq (1 - e^{-13})cdot tildeDelta(D)$ for large enough $tildeDelta(D)$." @default.
- W4291335965 created "2022-08-15" @default.
- W4291335965 creator A5089252105 @default.
- W4291335965 date "2014-09-26" @default.
- W4291335965 modified "2023-09-29" @default.
- W4291335965 title "The m-Degenerate Chromatic Number of a Digraph" @default.
- W4291335965 doi "https://doi.org/10.48550/arxiv.1409.7535" @default.
- W4291335965 hasPublicationYear "2014" @default.
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