H-coloring degree-bounded (acyclic) 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. 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. Interestingly, the situation changes when we further restrict the inputs to be acyclic digraphs. In two of the three cases the NP-complete problems remain NP-complete. In the third case, the H-coloring problem turns out to be polynomial time solvable for acyclic digraphs with in-degrees at most 2, regardless of the out-degree bound (respectively with out-degrees at most 2, regardless of the in-degree bound). We also show that in this case as soon as both in-and out-degrees are bounded by constants greater than or equal to 3, the H-coloring problem is once again NP-complete, even for acyclic digraphs. 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. It has been verified for H-coloring of digraphs, but the degree bounds are quite high, and are stated in terms of the sum of in-and out-degrees. Our results confirm the conjecture with the best possible bounds that are needed for NP-completeness; moreover, they underscore the fact that the bound must apply separately to both the indegrees and the out-degrees.