Antimagic Orientation of Forests

TL;DR

This paper investigates antimagic orientations of forests, proving that certain subdivided forests with at most one isolated vertex can be oriented to have antimagic labelings, supporting a broader conjecture.

Contribution

It proves that forests derived from subdividing edges of a given forest with at most one isolated vertex admit antimagic orientations.

Findings

Supports the conjecture that all connected graphs have antimagic orientations.

Proves that subdivided forests with specific conditions admit antimagic orientations.

Provides a new class of graphs confirmed to have antimagic orientations.

Abstract

An antimagic labeling of a digraph $D$ with $n$ vertices and $m$ arcs is a bijection from the set of arcs of $D$ to $\{1,2,\cdots,m\}$ such that all $n$ oriented vertex-sums are pairwise distinct, where the oriented vertex-sum of a vertex is the sum of labels of all arcs entering that vertex minus the sum of labels of all arcs leaving it. A graph $G$ admits an antimagic orientation if $G$ has an orientation $D$ such that $D$ has an antimagic labeling. Hefetz, M{\"{u}}tze and Schwartz conjectured every connected graph admits an antimagic orientation. In this paper, we support this conjecture by proving that any forest obtained from a given forest with at most one isolated vertex by subdividing each edge at least once admits an antimagic orientation.