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$\boldsymbol{\mathcal{CF}}$-CONNECTED GRAPHS FOR 
$\boldsymbol{K}_{\boldsymbol{m,n}}$Published online by Cambridge University Press: 01 December 2020
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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${K}_{5,n}$
’, J. Combin. Theory 9 (1970), 315–323.CrossRefGoogle ScholarFull text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.
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