D-neighborhood characterization of D-antimagic graphs

Characterize precisely the graphs that admit a $D$-antimagic labeling by proving or disproving that a graph is $D$-antimagic if and only if every vertex has a distinct $D$-neighborhood.

Background

The paper describes a generalization of distance-antimagic labeling in which the neighborhood of each vertex is replaced by its DD-neighborhood, consisting of vertices at directed or undirected distances belonging to a prescribed distance set DD.

The stated conjecture extends the neighborhood criterion from ordinary distance-antimagic graphs to arbitrary distance sets, asserting that distinct DD-neighborhoods are both necessary and sufficient for a DD-antimagic labeling.

References

Although much research has focused on antimagic properties in undirected graphs, the corresponding study for directed graphs, where the direction of arcs adds a layer of complexity, is still in its infancy. This paper aims to explore this less studied area and bridge that gap by extending the concept of $D$-antimagic labeling to oriented graphs, specifically in the context of linear forest graphs.

— D-Antimagic Labelings on Oriented Linear Forests  (2501.05035 - Abrar et al., 9 Jan 2025) in Conjecture 2, Section 1 (Introduction)