Anti-magic labeling of regular graphs

TL;DR

This paper proves that all regular graphs with even degree are antimagic, resolving an open problem and extending previous results on bipartite and odd degree regular graphs.

Contribution

It establishes that every even degree regular graph is antimagic, filling a significant gap in the understanding of antimagic labelings.

Findings

All even degree regular graphs are antimagic.

Confirms antimagic property for bipartite and odd degree regular graphs.

Completes the classification of regular graphs regarding antimagic labelings.

Abstract

A graph $G=(V,E)$ is antimagic if there is a one-to-one correspondence $f: E \to \{1,2,\ldots, |E|\}$ such that for any two vertices $u,v$, $\sum_{e \in E(u)}f(e) \ne \sum_{e\in E(v)}f(e)$. It is known that bipartite regular graphs are antimagic and non-bipartite regular graphs of odd degree at least three are antimagic. Whether all non-bipartite regular graphs of even degree are antimagic remained an open problem. In this paper, we solve this problem and prove that all even degree regular graphs are antimagic. The paper was submitted to December 2014 by Journal of Graph Theory.