The general position number of Cartesian products involving a factor with small diameter

•The gp-number is determined for the Cartesian product of a tree and a complete graph.•The gp-numbers are demonstrated for the Cartesian product of a complete graph with a cycle and that of a complete multipartite graph with a path, respectively.•An upper bound on the gp-number is proved for the Cartesian products of any two connected graphs G and H, and the bound is sharp if G and H have diameters at most 2. Moreover, the equality holds if and only if G and H are both generalized complete graphs. A vertex subset R of a graph G is called a general position set if any triple V0⊆R is non-geodesic, this is, the three elements of V0 do not lie on the same geodesic in G. The general position number (gp-number for short) gp(G) of G is the number of vertices in a largest general position set in G. In this paper we first determine some formulae for the gp-numbers of Cartesian products involving a complete graph and of the Cartesian product of a complete multipartite graph with a path, respectively. Moreover, it is proved that gp(G□H)≤n(G)+n(H)−2 for any Cartesian product G□H with equality holding if and only if G and H are both generalized complete graphs, that is, a special class of graphs with diameters at most 2. Finally several open problems are proposed on the gp-numbers of Cartesian products.