Caterpillars Have Antimagic Orientations

TL;DR

This paper proves that caterpillars, a subclass of trees, can be oriented to have antimagic labelings, supporting the conjecture that all connected graphs admit such orientations.

Contribution

It provides a constructive proof that caterpillars have antimagic orientations, advancing understanding of antimagic labelings in graph theory.

Findings

Caterpillars have antimagic orientations.

Constructive technique used for proof.

Supports the conjecture for broader classes of graphs.

Abstract

An antimagic labeling of a directed graph $D$ with $m$ arcs is a bijection from the set of arcs of $D$ to $\{1,\dots,m\}$ such that all oriented vertex sums of vertices in $D$ are pairwise distinct, where the oriented vertex sum of a vertex $u$ is the sum of labels of all arcs entering $u$ minus the sum of labels of all arcs leaving $u$. Hefetz, M\"utze, and Schwartz conjectured that every connected graph admits an antimagic orientation, where an antimagic orientation of a graph $G$ is an orientation of $G$ which has an antimagic labeling. We use a constructive technique to prove that caterpillars, a well-known subclass of trees, have antimagic orientations.