The complexity of the proper orientation number

• Published in: Information processing letters Vol. 113; no. 19-21; pp. 799 - 803
• Main Authors: Ahadi, A., Dehghan, A.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01-09-2013; Elsevier Sequoia S.A
• Subjects: Algorithms; Complexity; Computational complexity; Computer science; Data processing; Graph algorithms; Graph orientation; Graphs; Mathematical problems; NP-completeness; Numerical analysis; Orientation; Planar 3-SAT (type 2); Polynomial algorithms; Polynomials; Proper orientation; Studies; Vertex coloring; Graph orientation; Polynomial algorithms; Planar 3-SAT (type 2); Vertex coloring; NP-completeness; Proper orientation; Computational complexity
• ISSN: 0020-0190; 1872-6119
• DOI: 10.1016/j.ipl.2013.07.017

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, dD−(v)≠dD−(u) where dD−(v) is the number of edges with head v in D. The proper orientation number of G is defined as χ→(G)=minD∈Γmaxv∈V(G)dD−(v) where Γ is the set of proper orientations of G. We have χ(G)−1⩽χ→(G)⩽Δ(G), where χ(G) and Δ(G) denote the chromatic number and the maximum degree of G, respectively. 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.
•We study the computational complexity of the proper orientation number.•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.