Small H-Coloring Problems for Bounded Degree Digraphs

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.