On the proper orientation number of bipartite graphs

An orientation of a graph G is a digraph D obtained from G by replacing each edge by exactly one of the two possible arcs with the same endvertices. For each v ∈ V (G), the indegree of v in D, denoted by d− D (v), is the number of arcs with head v in D. An orientation D of G is proper if d− D (u) 6= d− D (v), for all uv ∈ E(G). The proper orientation number of a graph G, denoted by − →χ (G), is the minimum of the maximum indegree over all its proper orientations. It is well-known that − →χ (G) ≤ ∆(G), for every graph G. In this paper, we first prove that − →χ (G) ≤ ⌊( ∆(G) + √ ∆(G) ) /2 ⌋ + 1 if G is a bipartite graph, and − →χ (G) ≤ 4 if G is a tree. We then prove that deciding whether − →χ (G) ≤ ∆(G) − 1 is an NP-complete problem. We also show that it is NP-complete to decide whether − →χ (G) ≤ 2, for planar subcubic graphs G. Moreover, we prove that it is NP-complete to decide whether − →χ (G) ≤ 3, for planar bipartite graphs G with maximum degree 5.