ON PROBLEMS OF
-CONNECTED GRAPHS FOR
Keyword(s):
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 $ .
2013 ◽
Vol 22
(5)
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pp. 783-799
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1968 ◽
Vol 11
(5)
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pp. 729-732
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2013 ◽
Vol 3
(3)
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pp. 390-396
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2015 ◽
pp. 55-58
2021 ◽
Vol 12
(2)
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pp. 1040-1046