From the Crossing Numbers of K5 + Pn and K5 + Cn to the Crossing Numbers of Wm + Sn and Wm + Wn

Abstract

The crossing number of a graph is a significant measure that indicates the complexity of the graph and the difficulty of visualizing it. In this paper, we examine the crossing numbers of join products involving the complete graph K5 with discrete graphs, paths, and cycles. We analyze optimal drawings of K5, identify all five non-isomorphic drawings, and address previously hypothesized crossing numbers for K5+Pn, and K5+Cn. Through a simplified approach, we first establish cr(K5+Dn) and then extend our method to prove the crossing numbers cr(K5+Pn) and cr(K5+Cn). These results also lead to new hypotheses for cr(Wm+Sn) and cr(Wm+Wn) involving wheels and stars. Our findings correct previous inaccuracies in the literature and provide a foundation for future research.