There are several known exact results on the crossing numbers of Cartesian products of paths, cycles, and complete graphs. In this paper we prove that the crossing number of Cartesian product between a path on n vertices, P n and a special case of Petersen graph P (3, 1), P (3, 1) × P n is 4(n − 1) for n 1.

An m-prism is the Cartesian product of an m-cycle and a path with 2 vertices. We prove that the crossing number of the join of an m-prism (m ! 4) and a graph with k isolated vertices is km for each k 2 f1, 2g: We then use this result to prove that the crossing number of the Cartesian product of a 5-prism and a path with n vertices is 10ðn À 1Þ: This answers partially the conjecture raised by Peng and Yiew (in 2006) in the affirmative.