A connected graph G is $\mathcal {CF}$ -connected if there is a path between every pair of vertices with no crossing on its edges for each optimal drawing of G. We conjecture that a complete bipartite graph $K_{m,n}$ is $\mathcal {CF}$ -connected if and only if it does not contain a subgraph of $K_{3,6}$ or $K_{4,4}$ . We establish the validity of this conjecture for all complete bipartite graphs $K_{m,n}$ for any $m,n$ with $\min \{m,n\}\leq 6$ , and conditionally for $m,n\geq 7$ on the assumption of Zarankiewicz’s conjecture that $\mathrm {cr}(K_{m,n})=\big \lfloor \frac {m}{2} \big \rfloor \big \lfloor \frac {m-1}{2} \big \rfloor \big \lfloor \frac {n}{2} \big \rfloor \big \lfloor \frac {n-1}{2} \big \rfloor $ .

We give the crossing number of the join product $W_{4}+D_{n}$, where $W_{4}$ is the wheel on five vertices and $D_{n}$ consists of $n$ isolated vertices. The proof is based on calculating the minimum number of crossings between two different subgraphs from the set of subgraphs which do not cross the edges of the graph $W_{4}$ and from the set of subgraphs which cross the edges of $W_{4}$ exactly once.

The exact crossing number is only known for a small number of families of graphs. Many of the families for which crossing numbers have been determined correspond to cartesian products of two graphs. Here, the cartesian product of the sunlet graph, denoted ${\mathcal{S}}_{n}$, and the star graph, denoted $K_{1,m}$, is considered for the first time. It is proved that the crossing number of ${\mathcal{S}}_{n}\Box K_{1,2}$ is $n$, and the crossing number of ${\mathcal{S}}_{n}\Box K_{1,3}$ is $3n$. An upper bound for the crossing number of ${\mathcal{S}}_{n}\Box K_{1,m}$ is also given.

We extend known results concerning crossing numbers by giving the crossing number of the join product $G+D_{n}$, where the connected graph $G$ consists of one $4$-cycle and of two leaves incident with the same vertex of the $4$-cycle, and $D_{n}$ consists of $n$ isolated vertices. The proofs are done with the help of software that generates all cyclic permutations for a given number $k$ and creates a graph for calculating the distances between all $(k-1)!$ vertices of the graph.