Regular graphs are antimagic

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.
Comment: In a previous version of the paper, the authors gave a proof that regular graphs are antimagic. However, in the proof of Claim 6, case 2 assumes that $f(e)>\ell$ for every $e\in E(v_{i-1})-E'_i$. This assumption does not hold for edges in $E^\sigma_i$, thus the subsequent calculations are incorrect. The aim of the present erratum is to fix the proof