The crossing numbers of products of path with graphs of order six

The crossing numbers of Cartesian products of paths, cycles or stars with all graphs of order at most four are known. For the path Pn of length n, the crossing numbers of Cartesian products G Pn for all connected graphs G on five vertices are also known. In this paper, the crossing numbers of Cartesian products G Pn for graphs G of order six are studied. Let H denote the unique tree of order six with two vertices of degree three. The main contribution is that the crossing number of the Cartesian product H Pn is 2(n − 1). In addition, the crossing numbers of G Pn for fourty graphs G on six vertices are collected.