D-Antimagic Labelings of Oriented Star Forests

TL;DR

This paper characterizes $D$-antimagic labelings in oriented star forests, providing conditions for their existence and showing that such labelings are always possible for any $D$ with appropriate orientations.

Contribution

It offers a complete classification of $D$-antimagic labelings for oriented star forests, including necessary and sufficient conditions and existence proofs.

Findings

All orientations and $D$s for $D$-antimagic oriented stars are characterized.

Necessary and sufficient conditions for $D$-antimagic oriented star forests are established.

For any $D$, an orientation exists that admits a $D$-antimagic labeling.

Abstract

For a distance set $D$, an oriented graph $\overrightarrow{G}$ is $D$-antimagic if there exists a bijective vertex labeling such that the sum of all labels of $D$-out-neighbors is distinct for each vertex. This paper provides all orientations and all possible $D$s of a $D$-antimagic oriented star. We provide necessary and sufficient condition for $D$-antimagic oriented star forest containing isomorphic oriented stars. We show that for all possible $D$s, there exists an orientation for a star forest to admit a $D$-antimagic labeling.