Antimagic labelings of regular bipartite graphs: An application of the Marriage Theorem

TL;DR

This paper proves that all regular bipartite graphs with degree at least 2 are antimagic, using the Marriage Theorem to establish the existence of such labelings, thus supporting Ringel's conjecture for this class of graphs.

Contribution

It demonstrates that every regular bipartite graph with degree at least 2 is antimagic, applying the Marriage Theorem in a novel way for this class of graphs.

Findings

All regular bipartite graphs with degree ≥ 2 are antimagic.

The proof relies on the Marriage Theorem.

Supports Ringel's conjecture for bipartite graphs.

Abstract

A labeling of a graph is a bijection from $E(G)$ to the set $\{1, 2,..., |E(G)|\}$. A labeling is \textit{antimagic} if for any distinct vertices $u$ and $v$, the sum of the labels on edges incident to $u$ is different from the sum of the labels on edges incident to $v$. We say a graph is antimagic if it has an antimagic labeling. In 1990, Ringel conjectured that every connected graph other than $K_2$ is antimagic. In this paper, we show that every regular bipartite graph (with degree at least 2) is antimagic. Our technique relies heavily on the Marriage Theorem.