The complexity of the proper orientation number

Abstract

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.

On the proper orientation number of bipartite graphs

“…Ahadi and Dehgan [2] showed that it is NP-complete to decide whether − → χ (G) ≤ 2 for planar graphs G by using a reduction from the Planar 3-SAT problem. We first improve this result by showing that it is NP-complete to decide whether the proper orientation number of planar subcubic graphs is at most 2.…”

“…The proper orientation number of a graph G, denoted by − → χ (G), is the minimum integer k such that G admits a proper k-orientation. This graph parameter was introduced by Ahadi and Dehghan [2]. It is well-defined for any graph G since one can always obtain a proper ∆(G)-orientation (see [2]).…”

“…This graph parameter was introduced by Ahadi and Dehghan [2]. It is well-defined for any graph G since one can always obtain a proper ∆(G)-orientation (see [2]). In other words, − → χ (G) ≤ ∆(G).…”

Proper 3-orientations of bipartite planar graphs with minimum degree at least 3

“…The proper orientation number of G, denoted by − → χ (G), is the minimum integer k such that G admits a proper k-orientation. Proper orientation number is defined by Ahadi and Dehghan [1], and see the related research [2,3,6]. Knox et al [6] showed the following.…”

Proper orientation number of triangle‐free bridgeless outerplanar graphs

“…They observed that this parameter is well-defined for any graph G since one can always obtain a proper ∆(G)-orientation, where ∆(G) is the maximum degree of G. This fact can be proved by induction on the size of G by removing a vertex of maximum degree and orienting all edges towards this vertex. Every proper orientation of a graph G induces a proper vertex-coloring of G. Thus, ω(G) − 1 ≤ χ(G) − 1 ≤ χ(G) ≤ ∆(G), where ω(G) is the number of vertices in a maximum clique of G. Ahadi and Dehghan [1] proved that it is NP-complete to compute χ(G) even for planar graphs. Araujo et al [2] strengthened this result by showing it holds for bipartite planar graphs of maximum degree 5.…”