The Cartesian product of graphs was introduced more than 50 years ago and many fundamental results were obtained since then. Nevertheless, in the last years several basic problems on the Cartesian product were solved and interesting theorems proved on topics of contemporary interest in graph theory. Here we survey recent developments on the structure of the Cartesian product with emphasis on th...

This article proves the following result: Let G and G′ be graphs of orders n and n′, respectively. Let G∗ be obtained from G by adding to each vertex a set of n′ degree 1 neighbors. If G∗ has game coloring number m and G′ has acyclic chromatic number k, then the Cartesian product G G′ has game chromatic number at most k(k+m − 1). As a consequence, the Cartesian product of two forests has game c...

This paper is concerned with the recognition of approximate graph products with respect to the Cartesian product. Most graphs are prime, although they can have a rich product-like structure. The proposed algorithms are based on a local approach that covers a graph by small subgraphs, so-called partial star products, and then utilizes this information to derive the global factors and an embeddin...