A weighted orientation of a graph G is a pair (D, w) where D is an orientation of G and w is
an arc-weighting of D, that is an application A (D)→ N∖{0}. The in-weight of a vertex v in a …

An orientation D of a graph G=(V, E) is a digraph obtained from G by replacing each edge by
exactly one of the two possible arcs with the same end vertices. For each v∈ V (G), the …

An orientation of a graph G is in–out–proper if any two adjacent vertices have different in–
out-degrees, where the in–out-degree of each vertex is equal to the in-degree minus the out …

An orientation D of G is proper if for every xy∈ E (G), we have d D-(x)≠ d D-(y). An
orientation D is ap-orientation if the maximum in-degree of a vertex in D is at most p. The …

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 …

A weighted orientation of a graph G is a function (D, w) with an orientation D of G and with a
weight function w: E (G)→ Z+. The in-weight w D−(v) of a vertex v in D is the value Σ u∈ …

arXiv:2004.06964v1 [math.CO] 15 Apr 2020 Page 1 Note on weighted proper orientations of
outerplanar graphs Ruijuan Gu1, Gregory Gutin2, Yongtang Shi3, Zhenyu Taoqiu3 1 Sino-European …

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) …

In this thesis, we consider the following topics on directed graphs: proper orientation and
biorientation numbers of outerplanar graphs, strong arc-disjoint decompositions of …