The complexity of the proper orientation number

Authors:
A. Ahadi;A. Dehghan
Affiliations:
Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran;Department of Mathematics and Computer Science, Amirkabir University of Technology, Tehran, Iran
Venue:
Information Processing Letters
Year:
2013

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Abstract

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"D^-(v)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 @g-(G)=min"D"@?"@Cmax"v"@?"V"("G")d"D^-(v) where @C is the set of proper orientations of G. We have @g(G)-1=(G)=(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.