Compatible Hamilton cycles in Dirac graphs

Abstract

A graph is Hamiltonian if it contains a cycle passing through every vertex exactly once. A celebrated theorem of Dirac from 1952 asserts that every graph on n≥3 vertices with minimum degree at least n/2 is Hamiltonian. We refer to such graphs as Dirac graphs. In this paper we obtain the following strengthening of this result. Given a graph G=(V,E), an incompatibility system F over G is a family F = {F v } vV such that for every vV, the set F v is a family of unordered pairs F v ⊆ {{e,e′}}: ee′ ∈ E,ee′ = {v}}. An incompatibility system is Δ-bounded if for every vertex v and an edge e incident to v, there are at most Δ pairs in F v containing e. We say that a cycle C in G is compatible with F if every pair of incident edges e,e′ of C satisfies {e,e′}∉F v , where v=ee′. This notion is partly motivated by a concept of transition systems defined by Kotzig in 1968, and can be viewed as a quantitative measure of robustness of graph properties. We prove that there is a constant μ>0 such that for every μn-bounded incompatibility system F over a Dirac graph G, there exists a Hamilton cycle compatible with F. This settles in a very strong form a conjecture of Häggkvist from 1988.

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References

  1. [1]

    N. Alon and G. Gutin: Properly colored Hamilton cycles in edge-colored complete graphs, Random Struct. Algor. 11 (1997), 179–186.

    MathSciNet  Article  MATH  Google Scholar 

  2. [2]

    B. Bollobás and P. Erdős: Alternating hamiltonian cycles, Israel J. Math. 23 (1976), 126–131.

    MathSciNet  Article  MATH  Google Scholar 

  3. [3]

    J. Bondy: Paths and cycles, in: Handbook of Combinatorics, Vol. 1 (edited by R. Graham, M. Grotschel, and L. Lovász), Elsevier, Amsterdam (1995), 5–110.

    Google Scholar 

  4. [4]

    J. Böttcher, M. Schacht and A. Taraz: Proof of the bandwidth conjecture of Bollobás and Komlós, Mathematische Annalen 343 (2009), 175–205.

    MathSciNet  Article  MATH  Google Scholar 

  5. [5]

    C. Chen and D. Daykin: Graphs with Hamiltonian cycles having adjacent lines different colors, J. Combin. Theory B 21 (1976), 135–139.

    MathSciNet  Article  MATH  Google Scholar 

  6. [6]

    D. Christofides, D. Kühn and D. Osthus: Edge-disjoint Hamilton cycles in graphs, J. Combin. Theory B 102 (2012), 1035–1060.

    MathSciNet  Article  MATH  Google Scholar 

  7. [7]

    B. Csaba, D. Kühn, A. Lo, D. Osthus and A. Treglown: Proof of the 1-factorization and Hamilton decomposition conjectures, Mem. Amer. Math. Soc., to appear.

  8. [8]

    B. Cuckler and J. Kahn: Hamiltonian cycles in Dirac graphs, Combinatorica 29 (2009), 299–326.

    MathSciNet  Article  MATH  Google Scholar 

  9. [9]

    D. E. Daykin: Graphs with cycles having adjacent lines different colors, J. Combin. Theory B 20 (1976), 149–152.

    MathSciNet  Article  MATH  Google Scholar 

  10. [10]

    A. Ferber, M. Krivelevich and B.Sudakov: Counting and packing Hamilton cycles in dense graphs and oriented graph, J. Combin. Theory B, to appear.

  11. [11]

    P. Heinig: On prisms, Möbius ladders and the cycle space of dense graphs, European J. Combin. 36 (2014), 503–530.

    MathSciNet  Article  MATH  Google Scholar 

  12. [12]

    J. Komlós, G. Sárközy and E. Szemerédi: On the Pósa-Seymour conjecture, J. Graph Theory 29 (1998), 167–176.

    MathSciNet  Article  MATH  Google Scholar 

  13. [13]

    A. Kotzig: Moves without forbidden transitions in a graph, Matematicky časopis 18 (1968), 76–80.

    MathSciNet  MATH  Google Scholar 

  14. [14]

    M. Krivelevich, C. Lee and B. Sudakov: Robust Hamiltonicity of Dirac graphs, Trans. Amer. Math. Soc. 366 (2014), 3095–3130.

    MathSciNet  Article  MATH  Google Scholar 

  15. [15]

    M. Krivelevich, C. Lee and B. Sudakov: Compatible Hamilton cycles in random graphs, Random Struct. Algor., to appear.

  16. [16]

    D. Kühn and D. Osthus: Hamilton decompositions of regular expanders: a proof of Kelly's conjecture for large tournaments, Adv. Math. 237 (2013), 62–146.

    MathSciNet  Article  MATH  Google Scholar 

  17. [17]

    A. Lo: Properly coloured Hamiltonian cycles in edge-coloured complete graphs, Combinatorica, to appear.

  18. [18]

    L. Lovász: Combinatorial problems and exercises, 2nd ed., American Mathematical Soc., Providence (2007).

    MATH  Google Scholar 

  19. [19]

    L. Pósa: Hamiltonian circuits in random graphs, Discrete Math. 14 (1976), 359–364.

    MathSciNet  Article  MATH  Google Scholar 

  20. [20]

    J. Shearer: A property of the colored complete graph, Discrete Mathematics 25 (1979), 175–178.

    MathSciNet  Article  MATH  Google Scholar 

  21. [21]

    B. Sudakov and V. Vu: Local resilience of graphs, Random Struct. Algor. 33 (2008), 409–433.

    MathSciNet  Article  MATH  Google Scholar 

  22. [22]

    D. Woodall: A sufficient condition for Hamiltonian circuits, J. Combin. Theory B 25 (1978), 184–186.

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to Benny Sudakov.

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Research supported in part by USA-Israel BSF Grant 2010115 and by grant 912/12 from the Israel Science Foundation.

Research supported in part by NSF Grant DMS-1362326.

Research supported in part by SNSF grant 200021-149111 and by a USA-Israel BSF grant.

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Krivelevich, M., Lee, C. & Sudakov, B. Compatible Hamilton cycles in Dirac graphs. Combinatorica 37, 697–732 (2017). https://doi.org/10.1007/s00493-016-3328-7

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Mathematics Subject Classification (2010)

  • 05C45