Bipartite graphs with minimum degree at least 15 are antimagic

TL;DR

This paper proves that all bipartite graphs with a minimum degree of at least 15 are antimagic, advancing the understanding of the antimagic labeling conjecture for a broad class of graphs.

Contribution

It establishes that bipartite graphs with minimum degree at least 15 are antimagic, using novel combinatorial tools and labeling techniques.

Findings

Bipartite graphs with minimum degree ≥15 are proven to be antimagic.

Introduces a new labeling lemma ensuring divisibility properties of vertex sums.

Utilizes a combination of König's Theorem and subgraph analysis to achieve the result.

Abstract

An antimagic {labeling} of a graph $G=(V,E)$ is a one-to-one mapping $f: E\rightarrow\{1,2,\ldots,|E|\}$, ensuring that the vertex sums in $V$ are pairwise distinct, where a vertex sum of a vertex $v$ is defined as the sum of the labels of the edges incident to $v$. A graph is called antimagic if it admits an antimagic labeling. The Antimagic Labeling Conjecture, proposed by Hartsfield and Ringel in 1990, posits that every connected graph other than $K_2$ is antimagic. The conjecture was confirmed for graphs of average degree at least 4,182 in 2016 by Eccles, where it was stated that a similar approach could not reduce the bound below 1,000 from 4,182. This paper shows that every bipartite graph with minimum degree at least 15 is antimagic. Our approach relies on three tools: a consequence of K\"{o}nig's Theorem, the existence of a subgraph of a specific size that avoids Eulerian components, and a labeling lemma that ensures some vertex sums are divisible by three while others are not.