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 of G is a digraph obtained from G by replacing each edge by exactly one of
two possible arcs with the same endpoints. We call an orientation proper if neighboring …

Given a simple graph G, a weight function w: E (G)→ N∖{0}, and an orientation D of G, we
define μ−(D)= max v∈ V (G)⁡ w D−(v), where w D−(v)=∑ u∈ ND−(v) w (uv). We say that D …

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 …

A universal labeling of a graph G is a labeling of the edge set in G such that in every
orientation ℓ ℓ of G for every two adjacent vertices v and u, the sum of incoming edges of v …

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 …

Proper 3-orientations of bipartite planar graphs with minimum degree at least 3 - ScienceDirect
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A cactus is a connected graph in which each edge is contained in at most one cycle. We
generalize the concept of cactus graphs, ie, a k-cactus is a connected graph in which each …

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 …