D-Antimagic Labelings on Oriented Linear Forests (2501.05035v1)

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 $\omega_D(v)=\sum_{u\in N_D(v)}h(u)$, 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 $y$, $\omega_D(x)\ne \omega_D(y)$. 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.