Small h-coloring problems for bounded degree digraphs

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 in-degrees at most 1, regardless of the out-degree bound (respectively with out-degrees at most 1, regardless of the in-degree bound). On the other hand, as soon as both in-and out-degrees are bounded by constants greater than or equal to 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. Thus, our results verify the conjecture with very precise bounds on both in-and out-degrees that are needed for NP-completeness; in particular, the bounds underscore the fact that the sufficiently large bound must apply to both the in-degrees and the out-degrees. © 2013 Springer-Verlag Berlin Heidelberg.