Necessary and sufficient conditions for D-antimagic paths

Determine, for every distance set $D$, necessary and sufficient conditions under which an oriented path admits a $D$-antimagic labeling.

Background

The paper characterizes several classes of DD-antimagic oriented paths, including paths with 1∈D1\in D, paths with n−1∈Dn-1\in D, and paths with {0,n−2}⊆D\{0,n-2\}\subseteq D. These results cover important special cases and identify specific admissible orientations.

The authors explicitly leave the complete characterization for arbitrary distance sets unresolved.

References

It is then natural to ask the characterization of $D$-antimagic paths, for other sets of $D$. For any distance set $D$, find necessary and sufficient conditions for a path to admit a $D$-antimagic labeling.

— D-Antimagic Labelings on Oriented Linear Forests  (2501.05035 - Abrar et al., 9 Jan 2025) in Problem environment immediately following the final theorem of Section 2, Section 2 (D-Antimagic Labeling on Oriented Paths)