Conjectures about wheels without one edge

The~main aim of~the paper is to give the~crossing number of the~join product $G^\ast + D_n$ for the~graph $G^\ast$ isomorphic to 4-regular graph on six vertices except for two distinct edges with no common vertex such that two remaining vertices are still adjacent, and where $D_n$ consists of $n$ isolated vertices. The~proofs are done with the help of well-known exact values for crossing numbers of join products of four subgraphs $H_k$ of $G^\ast$ with discrete graphs. Further, we give a~conjecture concerning crossing numbers of the~join products of $D_n$ with $W_m\setminus e$ for both types edges $e$ of wheels $W_m$ of $m+1$ vertices.