A proper labeling of a graph is an assignment of integers to some elements of a graph, which may be the vertices, the edges, or both of them, such that we obtain a proper vertex coloring via the labeling subject to some conditions. The problem of proper labeling offers many variants and received a great interest during recent years. We consider the algorithmic complexity of some variants of the proper labeling problems, we present some polynomial time algorithms and $ \mathbf{NP} $-completeness results for them

Graph orientation is a well-studied area of graph theory. A proper orientation of a graph G = (V, E) is an orientation D of E(G) such that for every two adjacent vertices v and u, d −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.

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