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The complexity of the proper orientation number

Ahadi, A ; Sharif University of Technology | 2013

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  1. Type of Document: Article
  2. DOI: 10.1016/j.ipl.2013.07.017
  3. Publisher: 2013
  4. Abstract:
  5. 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)d D -(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
  6. Keywords:
  7. Computational complexity ; Graph orientation ; NP-completeness ; Planar 3-SAT (type 2) ; Polynomial algorithms ; Proper orientation ; Vertex coloring ; Graph orientations ; Polynomial algorithm ; Polynomial approximation ; Graph theory
  8. Source: Information Processing Letters ; Volume 113, Issue 19-21 , 2013 , Pages 799-803 ; 00200190 (ISSN)
  9. URL: http://www.sciencedirect.com/science/article/pii/S0020019013002068