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

Part of: Discrete mathematics in relation to computer science Algebraic combinatorics

Published online by Cambridge University Press:  01 December 2020

MICHAL STAŠ Open the ORCID record for MICHAL STAŠ [Opens in a new window]  and
JURAJ VALISKA Open the ORCID record for JURAJ VALISKA [Opens in a new window]
MICHAL STAŠ*
Affiliation:
Department of Mathematics and Theoretical Informatics, Faculty of Electrical Engineering and Informatics, Technical University of Košice, Letná 9, 042 00Košice, Slovak Republic
JURAJ VALISKA
Affiliation:
Department of Mathematics and Theoretical Informatics, Faculty of Electrical Engineering and Informatics, Technical University of Košice, Letná 9, 042 00 Košice, Slovak Republic e-mail: [email protected]
*
e-mail: [email protected]
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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 $ .

Keywords

complete bipartite graphcrossing numbercrossing sequenceoptimal drawing

MSC classification

Primary: 05E30: Association schemes, strongly regular graphs
Secondary: 68R10: Graph theory (including graph drawing)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 104 , Issue 2 , October 2021 , pp. 203 - 210
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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