The crossing numbers of join products of four graphs of order five with paths and cycles

. The crossing number cr( G ) of a graph G is the minimum number of edge crossings over all drawings of G in the plane. In the paper, we extend known results concerning crossing numbers of join products of four small graphs with paths and cycles. The crossing numbers of the join products G ∗ + P n and G ∗ + C n for the disconnected graph G ∗ consisting of the complete tripartite graph K 1 , 1 , 2 and one isolated vertex are given, where P n and C n are the path and the cycle on n vertices, respectively. In the paper also the crossing numbers of H ∗ + P n and H ∗ + C n are determined, where H ∗ is isomorphic to the complete tripartite graph K 1 , 1 , 3 . Finally, by adding new edges to the graphs G ∗ and H ∗ , we are able to obtain crossing numbers of join products of two other graphs G 1 and H 1 with paths and cycles.