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An approximate version of Jackson’s conjecture

  • Anita Liebenau (a1) and Yanitsa Pehova (a2)

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 2nn0 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 2nn0 vertices contains (1−ε)cn edge-disjoint Hamilton cycles.

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Corresponding author

*Corresponding author. Email: y.pehova@warwick.ac.uk

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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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References

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[1]Alon, N. and Spencer, J. H. (2016) The Probabilistic Method, fourth edition, Wiley.
[2]Alspach, B. (2008) The wonderful Walecki construction. Bull. Inst. Combin. Appl 52 52 720.
[3]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. 918, Springer.
[4]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.
[5]Dirac, G. A. (1952) Some theorems on abstract graphs. Proc. London Math. Soc. s3-2 6981.
[6]Ferber, A., Long, E. and Sudakov, B. (2018) Counting Hamilton decompositions of oriented graphs. Int. Math. Res. Not. 2018 69086933.
[7]Frieze, A. and Karoński, M. (2015) Introduction to Random Graphs, Cambridge University Press.
[8]Ghouila-Houri, A. (1960) Une condition suffisante d’existence d’un circuit Hamiltonien. CR Acad. Sci. Paris 251 495497.
[9]Häggkvist, R. (1993) Hamilton cycles in oriented graphs. Combin. Probab. Comput. 2 2532.
[10]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 19 510.
[11]Hilton, A. (1984) Hamiltonian decompositions of complete graphs. J. Combin. Theory Ser. B 36 125134.
[12]Jackson, B. (1981) Long paths and cycles in oriented graphs. J. Graph Theory 5 145157.
[13]Keevash, P., Kühn, D. and Osthus, D. (2009) An exact minimum degree condition for Hamilton cycles in oriented graphs. J. London Math. Soc. 79 144166.
[14]Kühn, D. and Osthus, D. (2013) Hamilton decompositions of regular expanders: a proof of Kelly’s conjecture for large tournaments. Adv. Math. 237 62146.
[15]Kühn, D. and Osthus, D. (2014) Hamilton decompositions of regular expanders: applications. J. Combin. Theory Ser. B 104 127.
[16]Kühn, D., Osthus, D. and Treglown, A. (2010) Hamilton decompositions of regular tournaments. Proc. London Math. Soc. 101 303335.
[17]Laskar, R. and Auerbach, B. (1976) On decomposition of r-partite graphs into edge-disjoint Hamilton circuits. Discrete Math. 14 265268.
[18]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. 813819, North-Holland.
[19]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. 157183, Academic Press.
[20]Ng, L. L. (1997) Hamiltonian decomposition of complete regular multipartite digraphs. Discrete Math. 177 279285.
[21]Ore, O. (1960) Note on Hamilton circuits. Amer. Math. Monthly 67 55.

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An approximate version of Jackson’s conjecture

  • Anita Liebenau (a1) and Yanitsa Pehova (a2)

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