D-Antimagic Labelings of Oriented 2-Regular Graphs

TL;DR

This paper studies $D$-antimagic labelings in oriented 2-regular graphs, providing characterizations for various cycle types and specific conditions on the distance set $D$.

Contribution

It offers new characterizations of $D$-antimagic labelings for different classes of oriented 2-regular graphs, including cycles and $ heta$-oriented graphs.

Findings

Characterization of $D$-antimagic oriented cycles for $|D|=1$

Characterization of $D$-antimagic unidirectional odd cycles for $|D|=2$

Characterization of $D$-antimagic $ heta$-oriented cycles and 2-regular graphs

Abstract

Given an oriented graph $\overrightarrow{G}$ and $D$ a distance set of $\overrightarrow{G}$, $\overrightarrow{G}$ is $D$-antimagic if there exists a bijective vertex labeling such that the sum of all labels of the $D$-out-neighbors of each vertex is distinct. This paper investigates $D$-antimagic labelings of 2-regular oriented graphs. We characterize $D$-antimagic oriented cycles, when $|D|=1$; $D$-antimagic unidirectional odd cycles, when $|D|=2$; and $D$-antimagic $\Theta$-oriented cycles. Finally, we characterize $D$-antimagic oriented 2-regular graphs, when $|D|=1$, and $D$-antimagic $\Theta$-oriented 2-regular graphs.