Regular graphs are antimagic

TL;DR

This paper discusses the antimagic labeling of regular graphs, correcting a previous proof that claimed all regular graphs are antimagic by fixing an invalid assumption in the original argument.

Contribution

The paper provides a corrected proof that regular graphs are antimagic, addressing errors in the previous version's argument.

Findings

Regular graphs are proven to be antimagic with the corrected proof.

The invalid assumption in the original proof is identified and rectified.

The result confirms the antimagic property for all regular graphs.

Abstract

An undirected simple graph $G=(V,E)$ is called antimagic if there exists an injective function $f:E\rightarrow\{1,\dots,|E|\}$ such that $\sum_{e\in E(u)} f(e)\neq\sum_{e\in E(v)} f(e)$ for any pair of different nodes $u,v\in V$. In a previous version of the paper, the authors gave a proof that regular graphs are antimagic. However, the proof of the main theorem is incorrect as one of the steps uses an invalid assumption. The aim of the present erratum is to fix the proof.