Antimagic Labelings of Forests

TL;DR

This paper proves that all forests with at most one degree-2 vertex are antimagic by providing an algorithmic approach to the zero-sum partition method, extending previous results on trees.

Contribution

It introduces an algorithmic representation of the zero-sum partition method and applies it to prove a broader class of forests are antimagic.

Findings

Forests with at most one degree-2 vertex are antimagic.

Algorithmic approach to zero-sum partition method developed.

Extends antimagic results from trees to certain forests.

Abstract

An antimagic labeling of a graph $G(V,E)$ is a bijection $f: E \to \{1,2, \dots, |E|\}$ so that $\sum_{e \in E(u)} f(e) \neq \sum_{e \in E(v)} f(e)$ holds for all $u, v \in V(G)$ with $u \neq v$, where $E(v)$ is the set of edges incident to $v$. We call $G$ antimagic if it admits an antimagic labeling. A forest is a graph without cycles; equivalently, every component of a forest is a tree. It was proved by Kaplan, Lev, and Roditty [2009], and by Liang, Wong, and Zhu [2014] that every tree with at most one vertex of degree-2 is antimagic. A major tool used in the proof is the zero-sum partition introduced by Kaplan, Lev, and Roditty [2009]. In this article, we provide an algorithmic representation for the zero-sum partition method and apply this method to show that every forest with at most one vertex of degree-2 is also antimagic.