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An approximate version of Jackson’s conjecture

Part of: Graph theory

Published online by Cambridge University Press:  30 June 2020

Anita Liebenau  and
Yanitsa Pehova
Anita Liebenau
Affiliation:
School of Mathematics and Statistics, UNSW Sydney, NSW 2052, Australia
Yanitsa Pehova*
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
*
*Corresponding author. Email: [email protected]
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Abstract

A diregular bipartite tournament is a balanced complete bipartite graph whose edges are oriented so that every vertex has the same in- and out-degree. In 1981 Jackson showed that a diregular bipartite tournament contains a Hamilton cycle, and conjectured that in fact its edge set can be partitioned into Hamilton cycles. We prove an approximate version of this conjecture: for every ε > 0 there exists n0 such that every diregular bipartite tournament on 2n ≥ n0 vertices contains a collection of (1/2–ε)n cycles of length at least (2–ε)n. Increasing the degree by a small proportion allows us to prove the existence of many Hamilton cycles: for every c > 1/2 and ε > 0 there exists n0 such that every cn-regular bipartite digraph on 2n ≥ n0 vertices contains (1−ε)cn edge-disjoint Hamilton cycles.

MSC classification

Primary: 05C20: Directed graphs (digraphs), tournaments
Secondary: 05C51: Graph designs and isomomorphic decomposition
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 29 , Issue 6 , November 2020 , pp. 886 - 899
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Footnotes

†

This work has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 648509). This publication reflects only its authors’ view; the European Research Council Executive Agency is not responsible for any use that may be made of the information it contains.

‡

Supported by the Australian research council (ARC), DE170100789 and DP180103684.

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