Cartesian Products of Regular Graphs are Antimagic

TL;DR

This paper proves that Cartesian products of regular graphs are antimagic, confirming the antimagic property for a broad class of graphs and extending previous results on specific cases like toroidal grids.

Contribution

It introduces a new class of antimagic graphs by proving Cartesian products of regular graphs are antimagic, expanding the scope of the antimagic graph conjecture.

Findings

Cartesian products of regular graphs are antimagic

All Cartesian products of two or more regular graphs are antimagic

Extends antimagic properties to a wide class of graph products

Abstract

An \emph{antimagic labeling} of a finite undirected simple graph with $m$ edges and $n$ vertices is a bijection from the set of edges to the integers $1,...,m$ such that all $n$ vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with the same vertex. A graph is called \emph{antimagic} if it has an antimagic labeling. In 1990, Hartsfield and Ringel \cite{HaRi} conjectured that every simple connected graph, but $K_2$, is antimagic. In this article, we prove that a new class of Cartesian product graphs are antimagic. In addition, by combining this result and the antimagicness result on toroidal grids (Cartesian products of two cycles) in \cite{Wan}, all Cartesian products of two or more regular graphs can be proved to be antimagic.