Regular graphs of odd degree are antimagic

TL;DR

This paper proves that all connected regular graphs with odd degree have an antimagic labeling, confirming a longstanding conjecture for this class of graphs.

Contribution

It establishes the antimagic labeling conjecture specifically for regular graphs of odd degree, a significant step in graph labeling theory.

Findings

Confirmed antimagic conjecture for regular odd degree graphs

Provided constructive proof for the labeling existence

Extended understanding of antimagic labelings in graph theory

Abstract

An antimagic labeling of a graph $G$ with $m$ edges is a bijection from $E(G)$ to $\{1,2,\ldots,m\}$ such that for all vertices $u$ and $v$, the sum of labels on edges incident to $u$ differs from that for edges incident to $v$. Hartsfield and Ringel conjectured that every connected graph other than the single edge $K_2$ has an antimagic labeling. We prove this conjecture for regular graphs of odd degree.