Dec 15, 2021 · In this work we study the proper arc labeling number and optimal proper arc labeling of directed graphs and we present some upper and lower bounds.
The proper arc labeling number of a directed graph G, denoted by χ p → ( G ) , is min ℓ ∈ Γ max v ∈ V ( G ) S ℓ ( v ) , where Γ is the set of proper arc ...
Dec 15, 2021 · An arc labeling ' of a directed graph G with positive integers is proper if for any two adjacent vertices v, u, we have S'ًvق 6¼ S'ًuق, where S'ًvق ...
Abstract: An arc labeling 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) , where ...
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In this work we study the proper arc labeling number of graphs and present some lower and upper bounds. We show that every directed graph G has a (not ...
May 4, 2020 · Define the weighted "length" of a directed path to be the maximum of all edge labels on that path (or 0 for a trivial path), and define the " ...
Dec 2, 2013 · Since D is simple, each vertex of D can have at most n−1 incoming arcs and n−1 outgoing arcs. Since each arc is directed toward one vertex and ...
A directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs.
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A Directed Acyclic Graph (DAG) is a digraph without any directed cycles. Lemma 1 If D is a DAG then D has at least one source (vertex of indegree 0) and at ...
Oct 22, 2024 · 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 ...