Oriented coloring in planar, bipartite, bounded degree 3 acyclic oriented graphs

Abstract

a b s t r a c tIt has been a challenging problem to determine the smallest graph class where a problem is proved to be hard. In the literature, this has been pointed out to be very important in order to establish the real nature of a combinatorial problem.An oriented k-coloring of an oriented graph ⃗ G = (V , ⃗ E) is a partition of V into k subsets such that there are no two adjacent vertices belonging to the same subset, and all the arcs between a pair of subsets have the same orientation. The decision problem k − oriented chromatic number (ocn k ) consists of an oriented graph ⃗ G and an integer k > 0, plus the question if there exists an oriented k-coloring of ⃗ G. By its strong appeal, many papers have presented NP-completeness proofs for ocn k . It was not known the complexity status of ocn k when the input graph ⃗ G satisfies that the underlying graph G has maximum degree 3.In this paper we prove that ocn 4 is NP-complete for an acyclic oriented graph ⃗ G such that G is at same time: connected, planar, bipartite, and with maximum degree 3.Our result defines a P versus NP-complete dichotomy with respect to the maximum degree ∆(G): ocn k is polynomial if ∆(G) < 3 and NP-complete if ∆(G) ≥ 3, since it is known that ocn 3 is in P, and that ocn k is in P when the underlying graph has ∆(G) ≤ 2.