D-Antimagic Labelings on Oriented Linear Forests

TL;DR

This work extends distance antimagic labeling to directed graphs by defining $D$-antimagic labelings for oriented graphs and exploring their existence on linear forest structures. It establishes a duality between a distance set $D$ and its complement $D^*$, and provides explicit constructive labelings for oriented paths, trees, and linear forests, including disjoint unions of paths. Key contributions include complete characterization for $P_n$ under several $D$ (notably cases with $\min(D)=1$ and the $\{0,n-2\}$-based regimes with special orientations $\Theta'$ and $\Theta''$), a rigidity result that distance antimagic trees are unidirectional paths, and practical labeling schemes for oriented linear forests with detailed constructions such as $\overrightarrow{mP_n}$ and the general $f_*$ mapping for $\{0,1\}$-antimagic forests. The results advance the theory of antimagic labelings in directed graphs and furnish concrete methods for labeling oriented linear forest classes, while posing open questions for broader $(D,\Omega)$ classifications.

Abstract

Let $\overrightarrow{G}$ be an oriented graph with the vertex set $V(\overrightarrow{G})$ and the arc set $A(\overrightarrow{G})$. Suppose that $D\subseteq \{0,1,\dots,\partial \}$ is a distance set where $\partial=\max \{d(u,v)<\infty|u,v\in V(\overrightarrow{G})\}$. Given a bijection $h:V(\overrightarrow{G}) \rightarrow\{1,2,\dots,|V(\overrightarrow{G})|\}$, the $D$-weight of a vertex $v\in V(\overrightarrow{G})$ is defined as $ω_D(v)=\sum_{u\in N_D(v)}h(u)$, where $N_D(v)=\{u\in V|d(v,u)\in D\}$. A bijection $h$ is called a $D$-antimagic labeling if for every pair of distinct vertices $x$ and $y$, $ω_D(x)\ne ω_D(y)$. An oriented graph $\overrightarrow{G}$ is called $D$-antimagic if it admits such a labeling. In addition to introducing the notion of $D$-antimagic labeling for oriented graphs, we investigate some properties of $D$-antimagic oriented graphs. In particular, we study $D$-antimagic linear forests for some $D$. We characterize $D$-antimagic paths where $1 \in D$, $n-1\in D$, or $\{0,n-2\}\subset D$. We characterize distance antimagic trees and forests. We conclude by constructing $D$-antimagic labelings on oriented linear forests.

Figures (4)

• Figure 1: A $\{0,2,3\}$-magic graph of order $5$ with magic constant $10$.
• Figure 2: The $\Theta'$ and $\Theta"$ orientations on a path.
• Figure 3: A {0,1}-antimagic labeling on $\Phi$ oriented $\overrightarrow{3P5}$.
• Figure 4: A $\{0,1\}$-antimagic labeling on $\mathcal{F}=2\overrightarrow{P_3}\cup \overrightarrow{P_5}\cup \overrightarrow{P_7}$. The black numbers are the labels and the red numbers inside the brackets are the $\{0,1\}$-weights.

Theorems & Definitions (45)

• Lemma 1.1: Handshake Lemma for Oriented Graphs
• Conjecture 1.1
• Conjecture 1.2
• Definition 1.1
• Theorem 2.1
• proof
• Corollary 2.1
• Theorem 2.2
• proof
• Theorem 2.3