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- W4297450003 abstract "An NP-complete coloring or homomorphism problem may become polynomial time solvable when restricted to graphs with degrees bounded by a small number, but remain NP-complete if the bound is higher. For instance, 3-colorability of graphs with degrees bounded by 3 can be decided by Brooks' theorem, while for graphs with degrees bounded by 4, the 3-colorability problem is NP-complete. We investigate an analogous phenomenon for digraphs, focusing on the three smallest digraphs H with NP-complete H-colorability problems. It turns out that in all three cases the H-coloring problem is polynomial time solvable for digraphs with degree bounds $Delta^{+} leq 1$, $Delta^{-} leq 2$ (or $Delta^{+} leq 2$, $Delta^{-} leq 1$). On the other hand with degree bounds $Delta^{+} leq 2$, $Delta^{-} leq 2$, all three problems are again NP-complete. A conjecture proposed for graphs H by Feder, Hell and Huang states that any variant of the $H$-coloring problem which is NP-complete without degree constraints is also NP-complete with degree constraints, provided the degree bounds are high enough. Our study is the first confirmation that the conjecture may also apply to digraphs." @default.
- W4297450003 created "2022-09-28" @default.
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- W4297450003 date "2012-11-27" @default.
- W4297450003 modified "2023-09-29" @default.
- W4297450003 title "Small H-coloring problems for bounded degree digraphs" @default.
- W4297450003 doi "https://doi.org/10.48550/arxiv.1211.6466" @default.
- W4297450003 hasPublicationYear "2012" @default.
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