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# ON PROBLEMS OF $\boldsymbol{\mathcal{CF}}$-CONNECTED GRAPHS FOR $\boldsymbol{K}_{\boldsymbol{m,n}}$

Published online by Cambridge University Press:  01 December 2020

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## Abstract

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 $\mathrm {cr}(K_{m,n})=\big \lfloor \frac {m}{2} \big \rfloor \big \lfloor \frac {m-1}{2} \big \rfloor \big \lfloor \frac {n}{2} \big \rfloor \big \lfloor \frac {n-1}{2} \big \rfloor$ .

## MSC classification

Type
Research Article
© 2020 Australian Mathematical Publishing Association Inc.

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## References

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ON PROBLEMS OF $\boldsymbol{\mathcal{CF}}$ -CONNECTED GRAPHS FOR $\boldsymbol{K}_{\boldsymbol{m,n}}$
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ON PROBLEMS OF $\boldsymbol{\mathcal{CF}}$ -CONNECTED GRAPHS FOR $\boldsymbol{K}_{\boldsymbol{m,n}}$
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