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 show that the Mallows measure on permutations of $1,\dots ,n$ arises as the law of the unique Gale–Shapley stable matching of the random bipartite graph with vertex set conditioned to be perfect, where preferences arise from the natural total ordering of the vertices of each gender but are restricted to the (random) edges of the graph. We extend this correspondence to infinite intervals, for which the situation is more intricate. We prove that almost surely, every stable matching of the random bipartite graph obtained by performing Bernoulli percolation on the complete bipartite graph $K_{{\mathbb Z},{\mathbb Z}}$ falls into one of two classes: a countable family $(\sigma _n)_{n\in {\mathbb Z}}$ of tame stable matchings, in which the length of the longest edge crossing k is $O(\log |k|)$ as $k\to \pm \infty $ , and an uncountable family of wild stable matchings, in which this length is $\exp \Omega (k)$ as $k\to +\infty $ . The tame stable matching $\sigma _n$ has the law of the Mallows permutation of ${\mathbb Z}$ (as constructed by Gnedin and Olshanski) composed with the shift $k\mapsto k+n$ . The permutation $\sigma _{n+1}$ dominates $\sigma _{n}$ pointwise, and the two permutations are related by a shift along a random strictly increasing sequence.

The thickness of a graph $G$ is the minimum number of planar subgraphs whose union is $G$. A $t$-minimal graph is a graph of thickness $t$ that contains no proper subgraph of thickness $t$. In this paper, upper and lower bounds are obtained for the thickness, $t\left( G\,\square \,H \right)$, of the Cartesian product of two graphs $G$ and $H$, in terms of the thickness $t\left( G \right)$ and $t\left( H \right)$. Furthermore, the thickness of the Cartesian product of two planar graphs and of a $t$-minimal graph and a planar graph are determined. By using a new planar decomposition of the complete bipartite graph ${{K}_{4k,\,4k}}$, the thickness of the Cartesian product of two complete bipartite graphs ${{K}_{n,n}}$ and ${{K}_{n,n}}$ is also given for $n\,\ne \,4k\,+\,1$.