  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.
