On the crossing numbers of join products of five graphs of order six with the discrete graph

The main purpose of this article is broaden known results concerning crossing numbers for join of graphs of order six. We give the crossing number of the join product G* + Dn, where the disconnected graph G* of order six consists of one isolated vertex and of one edge joining two nonadjacent vertices of the 5-cycle. In our proof, the idea of cyclic permutations and their combinatorial properties will be used. Finally, by adding new edges to the graph G*, the crossing numbers of Gi + Dn for four other graphs Gi of order six will be also established