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 degree bounds ∆ + ≤ 1, ∆ − ≤ 2 (or ∆ + ≤ 2, ∆ − ≤ 1). On the other hand with degree bounds ∆ + ≤ 2, ∆ − ≤ 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.

H-coloring degree-bounded (acyclic) digraphs

“…At the same time, we have to ensure that the color assigned to the literal vertex remains same in all copies, so that we have a consistent truth value assignment. Some of the proofs given below are more complex that the first proofs we gave in [12]. This is in order to allow an extension to the acyclic case discussed in the next section.…”

“…For the NP-complete cases, we consider adding the restriction that the input digraph is acyclic. This has arisen from the conference version of this paper [12] where we noticed that many of the tricks in the NP-completeness constructions required directed cycles. Surprisingly, we were able to duplicate almost all of the constructions while at the same time avoiding directed cycles, so these results also apply to acyclic degree-bounded digraphs.…”