The proper orientation number χ→(G) of a graph G is the minimum k such that there exists
an orientation of the edges of G with all vertex-outdegrees at most k and such that for any …

An orientation D of a graph G is a digraph obtained from G by replacing each edge by
exactly one of the two possible arcs with the same ends. An orientation D of a graph G is a k …

An arc labeling ℓ ℓ of a directed graph G with positive integers is proper if for any two
adjacent vertices v, u, we have S_ ℓ (v) ≠ S_ ℓ (u) S ℓ (v)≠ S ℓ (u), where S_ ℓ (v) S ℓ (v) …

Given an orientation D of the edges of a simple graph G, the indegree of a vertex v∈ V (G), d
D (v), is the number of arcs with head in v. Such orientation induces a coloring φ (v)= d D …

Uma orientação $ D $ de um grafo simples $ G=(V, E) $ é própria se os vértices~ $ u $ e~ $
v $ possuem graus de entrada distintos, sempre que~ $ uv\in E $. O número de orientação …