The Crossing Numbers of Join Products of Paths and Cycles with Four Graphs of Order Five

The main aim of the paper is to establish the crossing numbers of the join products of the paths and the cycles on n vertices with a connected graph on five vertices isomorphic to the graph K-1,K-1,K-3 / e obtained by removing one edge e incident with some vertex of order two from the complete tripartite graph K-1,K-1,K-3. The proofs are done with the help of well-known exact values for the crossing numbers of the join products of subgraphs of the considered graph with paths and cycles. Finally, by adding some edges to the graph under consideration, we obtain the crossing numbers of the join products of other graphs with the paths and the cycles on n vertices.

Automated summary

The paper aims to establish the crossing numbers of join products of paths and cycles on n vertices with a connected graph on five vertices, including adding some edges to determine the crossing numbers of join products of other graphs with paths and cycles on n vertices.