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An Ore-Type Theorem on Hamiltonian Square Cycles

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Abstract

The kth power of a cycle C is the graph obtained from C by joining every pair of vertices with distance at most k on C. The second power of a cycle is called a square cycle. Pósa conjectured that every graph with minimum degree at least 2n/3 contains a hamiltonian square cycle. Later, Seymour proposed a more general conjecture that if G is a graph with minimum degree at least (kn)/(k + 1), then G contains the kth power of a hamiltonian cycle. Here we prove an Ore-type version of Pósa’s conjecture that if G is a graph in which deg(u) + deg(v) ≥ 4n/3 − 1/3 for all non-adjacent vertices u and v, then for sufficiently large n, G contains a hamiltonian square cycle unless its minimum degree is exactly n/3 + 2 or n/3 + 5/3. A consequence of this result is an Ore-type analogue of a theorem of Aigner and Brandt.

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References

  1. Aigner M., Brandt S.: Embedding arbitrary graphs of maximum degree two. J. Lond. Math. Soc. 48, 39–51 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  2. Châu, P.: An Ore-type version of Pósa’s conjecture (2009, manuscript)

  3. Châu P., DeBiasio L., Kierstead H.A.: Pósa’s Conjecture for graphs of order at least 2 × 108. Random Struct. Algorithms 39(4), 507–525 (2011)

    Article  MATH  Google Scholar 

  4. Dirac G.A.: Some theorems on abstract graphs. Proc. Lond. Math. Soc. 2, 68–81 (1952)

    MathSciNet  Google Scholar 

  5. Erdös P.: Problem 9, Theory of Graphs and its Applications (M. Fielder ed.), Czech. Acad. Sci. Publ., Prague, p. 159 (1964)

  6. Fan G.H., Häggkvist R.: The square of a hamiltonian cycle. SIAM J. Discrete Math. 7(2), 203–212 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fan G.H., Kierstead H.A.: The square of paths and cycles. J. Comb. Theory Ser. B 63(1), 55–64 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fan G.H., Kierstead H.A.: Hamiltonian square paths. J. Comb. Theory Ser. B 67(2), 167–182 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fan G.H., Kierstead H.A.: Partitioning a graph into two square-cycles. J. Graph Theory 23(3), 241–256 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  10. Faudree, R.J., Gould, R.J., Jacobson, M.S., Schelp, R.: Seymour’s Conjecture. Advances in Graph Theory. In: Kulli, V.R. (ed.) Vishwa, Gulbarga, pp. 163–171 (1991)

  11. Hajnal, A., Szemerédi, E.: Proof of a conjecture of P. Erdös. Combinatorial Theory and its Applications, pp. 601–623. North-Holland (1970)

  12. Kierstead H.A., Kostochka A.V.: A short proof of the Hajnal–Szemerédi Theorem on equitable coloring. Comb. Prob. Comput. 17(2), 265–270 (2008)

    MathSciNet  MATH  Google Scholar 

  13. Kierstead H.A., Kostochka A.V.: An Ore-type theorem on equitable coloring. J. Comb. Theory Ser. B 98(1), 226–234 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Komlós J., Sárközy G.N., Szemerédi E.: Blow-up lemma. Combinatorica 17, 109–123 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  15. Komlós J., Sárközy G.N., Szemerédi E.: On the square of a hamiltonian cycle in dense graphs. Random Struct Algorithms 9(1-2), 193–211 (1996)

    Article  MATH  Google Scholar 

  16. Komlós J., Sárközy G.N., Szemerédi E.: Proof of the Seymour conjecture for large graphs. Ann. Comb. 2(1), 43–60 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  17. Komlós J., Simonovits M.: Szemerédi’s regularity lemma and its applications in graph theory. Combinatorics: Paul Erdös is eighty 2, 295–352 (1996)

    Google Scholar 

  18. Kostochka, A.V., Yu, G.: Extremal problems on packing of graphs. Oberwolfach reports, No 1, pp. 55–57 (2006)

  19. Kostochka A.V., Yu G.: Ore-type graph packing problems. Comb. Prob. Comput. 16(1), 167–169 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kostochka, A.V., Yu, G.: Graphs containing every 2-factor. Graphs and Combinatorics (to appear)

  21. Kühn D., Osthus D., Treglown A.: An Ore-type theorem for perfect packings in graphs. SIAM J. Discrete Math. 23(3), 1335–1355 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Levitt I., Sárközy G.N., Szemerédi E.: How to avoid using the Regularity Lemma: Pósa’s conjecture revisited. Discrete Math. 310(3), 610–641 (2010)

    Article  Google Scholar 

  23. Ore O.: Note on Hamilton circuits. Am. Math. Mon. 67, 55 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  24. Seymour, P.: Problem section in Combinatorics. In: McDonough, T.P., Mavron, V.C. (eds.) Proceedings of the British Combinatorial Conference 1973, pp. 201–202. Cambridge University Press, Cambridge (1974)

  25. Szemerédi, E.: Regular partitions of graphs. Colloques Internationaux C.N.R.S. No. 260-Problemes Combinatoires et Theorie des Graphes, Orsay, pp. 399–401 (1976)

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  1. Glendale Community College, Glendale, AZ, USA

    Phong Châu

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  1. Phong Châu
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Châu, P. An Ore-Type Theorem on Hamiltonian Square Cycles. Graphs and Combinatorics 29, 795–834 (2013). https://doi.org/10.1007/s00373-012-1161-3

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  • DOI: https://doi.org/10.1007/s00373-012-1161-3

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