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

The crossing number $ \text{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.