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 $ .

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关于连通图的问题

连通图G是$\数学{CF}$- 如果每对顶点之间有一条路径，并且对于每个最优绘图，其边上没有交叉点，则为连接G. 我们猜想一个完整的二分图$K_{m,n}$是$\数学{CF}$-连接当且仅当它不包含$K_{3,6}$要么$K_{4,4}$. 我们为所有完全二部图建立了这个猜想的有效性$K_{m,n}$对于任何$m,n$和$\min \{m,n\}\leq 6$，并且有条件地为$m,n\geq 7$基于 Zarankiewicz 猜想的假设$\mathrm {cr}(K_{m,n})=\big \lfloor \frac {m}{2} \big \rfloor \big \lfloor \frac {m-1}{2} \big \rfloor \大 \lfloor \frac {n}{2} \big \rfloor \big \lfloor \frac {n-1}{2} \big \rfloor $.