The crossing number of

The exact crossing number is known only for a few specific families of graphs. Cartesian products of two graphs belong to the first families of graphs for which the crossing number has been studied. Let P n be a path with n + 1 vertices. P n k , the k -power of the graph P n , is a graph on the same vertex set as P n and the edges that join two vertices of P n if the distance between them is at most k . Very recently, some results concerning crossing numbers of P n k were obtained. In this paper, the crossing numbers of the Cartesian product of P n 2 with the cycle C m are studied. It is proved that the crossing number of the graph P n 2 □ C 3 is 3 n − 3 , and the upper bound for the crossing number of the graph P n 2 □ C m is given.