D-Antimagic Labelings of Oriented 2-Regular Graphs

TL;DR

This work investigates $D$-antimagic labelings for oriented $2$-regular graphs, focusing on cycles and their orientations. By analyzing unidirectional and $Θ$-oriented cycles, it provides complete characterizations for $|D|=1$ and substantial results for $|D|=2$, including explicit constructions for odd unidirectional cycles and several $Θ$-oriented labelings, and extends these insights to multi-cycle $2$-regular graphs. For $|D|=1$, it fully characterizes when a cycle or disjoint union of cycles is $D$-antimagic, while for $|D| eq1$ it offers constructions and identifies open problems, notably for even cycles and larger distance sets. The paper also delivers a constructive framework, via labelings like $h_*$ and $f_*$, to realize $D$-antimagic labelings in $Θ$-oriented components and delineates when such labelings are possible in unions of cycles. Overall, it advances understanding of distance-based antimagic properties in oriented graphs and raises natural questions for broader distance sets and cycle orientations.

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 $Θ$-oriented cycles. Finally, we characterize $D$-antimagic oriented 2-regular graphs, when $|D|=1$, and $D$-antimagic $Θ$-oriented 2-regular graphs.

Figures (8)

• Figure 1: (a) $\{0,1,3,4\}$- and (b) $\{1,3,4,6\}$-antimagic labelings of $\Theta$ oriented of $\overrightarrow{C_{10}}$.
• Figure 2: $\{0,1\}$-antimagic labelings of unidirectional $\overrightarrow{C_7}$ and $\overrightarrow{C_8}$.
• Figure 3: $\{0,2\}$-antimagic labelings of unidirectional $\overrightarrow{C_7}$ and $\overrightarrow{{C_8}}$.
• Figure 4: $\{0,3\}$-antimagic labelings of $\overrightarrow{C_8}, \overrightarrow{C_{10}}$, and $\overrightarrow{C_{12}}$.
• Figure 5: $\{0,k\}$-antimagic labelings of unidirectional $\overrightarrow{C_7}$ where (a) $k=4$ and (b) $k=3$.

Theorems & Definitions (37)

• Definition 1.1
• Lemma 1.1: Handshake Lemma for Oriented Graphs
• Lemma 1.2
• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Lemma 2.2
• proof
• Theorem 2.1
• proof