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 ) ? d D - ( v ) , for all u v ? E ( G ) . The proper orientation number of a graph G, denoted by ? ? ( G ) , is the minimum of the maximum indegree over all its proper orientations. In this paper, we prove that ? ? ( G ) ? ( Δ ( G ) + Δ ( G ) ) / 2 + 1 if G is a bipartite graph, and ? ? ( G ) ? 4 if G is a tree. It is well-known that ? ? ( G ) ? Δ ( G ) , for every graph G. However, we prove that deciding whether ? ? ( G ) ? Δ ( G ) - 1 is already an NP -complete problem on graphs with Δ ( G ) = k , for every k ? 3 . We also show that it is NP -complete to decide whether ? ? ( G ) ? 2 , for planar subcubic graphs G. Moreover, we prove that it is NP -complete to decide whether ? ? ( G ) ? 3 , for planar bipartite graphs G with maximum degree 5.
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