Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-28T07:22:03.187Z Has data issue: false hasContentIssue false
  • English
  • Français

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]
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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.

References

Alon, N. and Spencer, J. H. (2016) The Probabilistic Method, fourth edition, Wiley.Google Scholar
Alspach, B. (2008) The wonderful Walecki construction. Bull. Inst. Combin. Appl 52 52 7–20.Google Scholar
Alspach, B., Bermond, J.-C. and Sotteau, D. (1990) Decomposition into cycles I: Hamilton decompositions. In Cycles and Rays, Vol. 301 of NATO ASI Series, pp. 9–18, Springer.Google Scholar
Csaba, B., Kühn, D., Lo, A., Osthus, D. and Treglown, A. (2016) Proof of the 1-factorization and Hamilton decomposition conjectures, Vol. 244, No. 1154 of Memoirs of the American Mathematical Society, AMS.CrossRefGoogle Scholar
Dirac, G. A. (1952) Some theorems on abstract graphs. Proc. London Math. Soc. s3-2 69–81.CrossRefGoogle Scholar
Ferber, A., Long, E. and Sudakov, B. (2018) Counting Hamilton decompositions of oriented graphs. Int. Math. Res. Not. 2018 6908–6933.CrossRefGoogle Scholar
Frieze, A. and Karoński, M. (2015) Introduction to Random Graphs, Cambridge University Press.Google Scholar
Ghouila-Houri, A. (1960) Une condition suffisante d’existence d’un circuit Hamiltonien. CR Acad. Sci. Paris 251 495–497.Google Scholar
Häggkvist, R. (1993) Hamilton cycles in oriented graphs. Combin. Probab. Comput. 2 25–32.CrossRefGoogle Scholar
Hetyei, G. (1975) On the 1-factors and Hamilton circuits of complete n-colorable graphs. Acta Acad. Paedagog. Civitate Pecs Ser. 6 Math. Phys. Chem. Tech 19 5–10.Google Scholar
Hilton, A. (1984) Hamiltonian decompositions of complete graphs. J. Combin. Theory Ser. B 36 125–134.CrossRefGoogle Scholar
Jackson, B. (1981) Long paths and cycles in oriented graphs. J. Graph Theory 5 145–157.CrossRefGoogle Scholar
Keevash, P., Kühn, D. and Osthus, D. (2009) An exact minimum degree condition for Hamilton cycles in oriented graphs. J. London Math. Soc. 79 144–166.CrossRefGoogle Scholar
Kühn, D. and Osthus, D. (2013) Hamilton decompositions of regular expanders: a proof of Kelly’s conjecture for large tournaments. Adv. Math. 237 62–146.CrossRefGoogle Scholar
Kühn, D. and Osthus, D. (2014) Hamilton decompositions of regular expanders: applications. J. Combin. Theory Ser. B 104 1–27.CrossRefGoogle Scholar
Kühn, D., Osthus, D. and Treglown, A. (2010) Hamilton decompositions of regular tournaments. Proc. London Math. Soc. 101 303–335.Google Scholar
Laskar, R. and Auerbach, B. (1976) On decomposition of r-partite graphs into edge-disjoint Hamilton circuits. Discrete Math. 14 265–268.CrossRefGoogle Scholar
Nash-Williams, C. St J. A. (1970) Hamiltonian lines in graphs whose vertices have sufficiently large valencies. In Combinatorial Theory and its Applications III (Proc. Colloq., Balatonfüred, 1969), pp. 813–819, North-Holland.Google Scholar
Nash-Williams, C. St J. A. (1971) Edge-disjoint Hamiltonian circuits in graphs with vertices of large valency. In Studies in Pure Mathematics: Papers Presented to Richard Rado (Mirsky, L., ed.), pp. 157–183, Academic Press.Google Scholar
Ng, L. L. (1997) Hamiltonian decomposition of complete regular multipartite digraphs. Discrete Math. 177 279–285.CrossRefGoogle Scholar
Ore, O. (1960) Note on Hamilton circuits. Amer. Math. Monthly 67 55.CrossRefGoogle Scholar