Construction of Antimagic Labeling for the Cartesian Product of Regular Graphs

Abstract

An antimagic labeling of a graph with p vertices and q edges is a bijection from the set of edges to the set of integers {1, 2, . . . , q} such that all vertex weights are pairwise distinct, where a vertex weight is the sum of labels of all edges incident with the vertex. A graph is antimagic if it has an antimagic labeling. In 1990, Hartsfield and Ringel conjectured that that every connected graph, except K ₂, is antimagic. Recently, using completely separating systems, Phanalasy et al. showed that for each ${k\geq 2,\,q\geq\binom{k+1}{2}}$ with k|2q, there exists an antimagic k-regular graph with q edges and p = 2q/k vertices. In this paper we prove constructively that certain families of Cartesian products of regular graphs are antimagic.