[Paper Review] Cartesian Products of Regular Graphs are Antimagic

Yongxi Cheng|ArXiv.org|Feb 14, 2006

Graph Labeling and Dimension Problems6 references3 citations

TL;DR

This paper proves that Cartesian products of regular graphs are antimagic, meaning they admit a labeling of edges with distinct integers 1 to m such that all vertex sums (sum of incident edge labels) are unique. The authors establish this by constructing such labelings for regular graph products and extend the result to all Cartesian products of two or more regular graphs, resolving a key case of the Hartsfield-Ringle antimagic conjecture.

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.

Motivation & Objective

• To prove that Cartesian products of regular graphs are antimagic, i.e., admit an edge labeling with distinct labels 1 to m such that all vertex sums are distinct.
• To extend the antimagic property to all Cartesian products of two or more regular graphs, building on prior results for toroidal grids.
• To provide a constructive method for generating antimagic labelings in regular graph products, contributing to the broader Hartsfield-Ringle conjecture.
• To resolve a specific class of graphs within the long-standing antimagic labeling conjecture, offering a general framework for future extensions.

Proposed method

• Constructs an antimagic labeling for Cartesian products of regular graphs using systematic edge labeling techniques.
• Employs combinatorial arguments to ensure that vertex sums—defined as the sum of labels on incident edges—are pairwise distinct.
• Leverages known results on toroidal grids (Cartesian products of two cycles) as a foundational case for the general proof.
• Applies induction and symmetry arguments to extend the labeling construction to products of more than two regular graphs.
• Uses the regularity of the graphs (constant degree) as a key structural property enabling uniform labeling patterns.
• Ensures bijectivity of the edge labeling to integers 1 through m, the number of edges, while maintaining distinct vertex sums.

Experimental results

Research questions

• RQ1 Can Cartesian products of regular graphs be labeled with distinct edge labels such that all vertex sums are unique?
• RQ2 Does the antimagic property extend from toroidal grids (products of two cycles) to all Cartesian products of two or more regular graphs?
• RQ3 What structural properties of regular graphs enable the construction of antimagic labelings in their Cartesian products?
• RQ4 Is there a general labeling method applicable to all regular graph products that guarantees distinct vertex sums?
• RQ5 How does the regularity of the factor graphs influence the feasibility and construction of antimagic labelings?

Key findings

• All Cartesian products of two or more regular graphs are antimagic, confirming the antimagic property for this broad class of graphs.
• The proof establishes a constructive method for generating antimagic labelings in such products, ensuring all vertex sums are pairwise distinct.
• The result generalizes prior findings on toroidal grids, which are Cartesian products of two cycles, to all regular graph products.
• The antimagic labeling is achieved via a bijective assignment of edge labels 1 to m such that no two vertices have the same sum of incident edge labels.
• The regularity of the graphs ensures sufficient symmetry and uniformity to allow consistent labeling patterns that avoid sum collisions.
• The work provides a significant step toward resolving the Hartsfield-Ringle conjecture, which posits that all simple connected graphs except K₂ are antimagic.