The Complexity of the Proper Orientation Number

Graph orientation is a well-studied area of graph theory. A proper orientation of a graph G = (V,E) is an orientationD of E(G) such that for every two adjacent vertices v and u, d D (v) 6= d D (u) where d D (v) is the number of edges with head v in D. The proper orientation number of G is defined as −→χ (G) = min D∈Γ max v∈V (G) d D (v) where Γ is the set of proper orientations of G. We have χ(G) − 1 ≤ −→χ (G) ≤ ∆(G). We show that, it is NP-complete to decide whether −→χ (G) = 2, for a given planar graph G. Also, we prove that there is a polynomial time algorithm for determining the proper orientation number of 3-regular graphs. In sharp contrast, we will prove that this problem is NP-hard for 4-regular graphs.