Classification of D-antimagic orientations of linear forests

Determine all distance sets $D$ and orientations $\Omega$ for which the oriented linear forest $\mathcal{F}$ is $D$-antimagic.

Background

The paper constructs a {0,1}\{0,1\}-antimagic labeling for every unidirectionally oriented linear forest, including forests whose components are arbitrary paths. It also establishes several results for disjoint unions of isomorphic paths and selected distance sets.

The authors explicitly identify the broader classification of all compatible distance sets and orientations for a general linear forest as an open problem.

References

Theorem \ref{thm:F} provides a $D$-antimagic labeling on unidirectional $\mathcal{F}$ where $D={0,1}$. Therefore, it is natural to propose the following open problem. Find all sets $D$ and orientation $\Omega$ such that $\mathcal{F}$ with orientation $\Omega$ is $D$-antimagic.

— D-Antimagic Labelings on Oriented Linear Forests  (2501.05035 - Abrar et al., 9 Jan 2025) in Problem environment at the end of Section 4, Section 4 (D-Antimagic Labeling on Oriented Linear Forests)