A new class of antimagic Cartesian product graphs

An 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 antimagic if it has an antimagic labeling. In 1990, Hartsfield and Ringel [5] conjectured that every simple connected graph, but K2, is antimagic. In this article, we prove that a new class of Cartesian product graphs are antimagic. In particular, by combining this result and the antimagicness result on toroidal grids (Cartesian products of two cycles) in [7], all Cartesian products of two or more regular graphs of positive degree can be proved to be antimagic.