Skip to main content
SpringerLink
Log in

A Cyclic Solution for an Infinite Class of Hamilton–Waterloo Problems

  • Original Paper
  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

The main result of this paper is the explicit construction, for any positive integer n, of a cyclic two-factorization of \(K_{50n+5}\) with \(20n+2\) two-factors consisting of five \((10n+1)\)-cycles and each of the remaining two-factors consisting of all pentagons. Then, applying suitable composition constructions, we obtain a few other two-factorizations also having two-factors of two distinct types.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Abel, R.J.R., Ge, G., Yin, J.: Resolvable and near-resolvable designs. In: Colbourn, C.J., Dinitz, J.H. (eds.) The CRC Handbook of Combinatorial Designs, 2nd edn, pp. 740–754. CRC Press, Boca Raton (2007)

    Google Scholar 

  2. Abel, R.J.R., Colbourn, C.J., Dinitz, J.H.: Mutually orthogonal latin squares (MOLS). In: Colbourn, C.J., Dinitz, J.H. (eds.) The CRC Handbook of Combinatorial Designs, 2nd edn, pp. 740–754. CRC Press, Boca Raton (2007)

    Google Scholar 

  3. Adams, P., Bryant, D.: Two-factorisations of complete graphs of orders fifteen and seventeen. Australas. J. Combin. 35, 113–118 (2006)

    MathSciNet  MATH  Google Scholar 

  4. Adams, P., Billington, E.J., Bryant, D., El-Zanati, S.I.: On the Hamilton–Waterloo problem. Graphs Combin. 18, 31–51 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Anderson, L.D.: Factorizations of graphs. In: Colbourn, C.J., Dinitz, J.H. (eds.) The CRC Handbook of Combinatorial Designs, 2nd edn, pp. 740–754. CRC Press, Boca Raton (2007)

    Google Scholar 

  6. Bryant, D., Danziger, P., Pettersson, W.: Bipartite \(2\)-factorizations of complete multipartite graphs. J. Graph Theory 78(4), 287–294 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bonvicini, S., Buratti, M.: Sharply vertex transitive 2-factorizations of Cayley graphs (preprint)

  8. Bryant, D., Danziger, P.: On bipartite \(2\)-factorizations of \(K_n - I\) and the Oberwolfach problem. J. Graph Theory 68, 22–37 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bryant, D., Rodger, C.: Cycle decompositions. In: Colbourn, C.J., Dinitz, J.H. (eds.) The CRC Handbook of Combinatorial Designs, 2nd edn, pp. 373–382. CRC Press, Boca Raton (2007)

    Google Scholar 

  10. Bryant, D., Scharaschkin, V.: Complete solutions to the Oberwolfach problem for an infinite set of orders. J. Combin. Theory Ser. B 99, 904–918 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Buratti, M., Del Fra, A.: Cyclic Hamiltonian cycle systems of the complete graph. Discrete Math. 279, 107–119 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Buratti, M., Rania, F., Zuanni, F.: Some constructions for cyclic perfect cycle systems. Discrete Math. 299, 33–48 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Buratti, M., Rinaldi, G.: On sharply vertex transitive \(2\)-factorizations of the complete graph. J. Combin. Theory Ser. A 111, 245–256 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Buratti, M., Rinaldi, G.: \(1\)-rotational \(k\)-factorizations of the complete graph and new solutions to the Oberwolfach problem. J. Combin. Des. 16, 87–100 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Danziger, P., Quattrocchi, G., Stevens, B.: The Hamilton-Waterloo problem for cycle sizes 3 and 4. J. Combin. Des. 17, 342–352 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Deza, A., Franek, F., Hua, W., Meszka, M., Rosa, A.: Solutions to the Oberwolfach problem for orders \(18\) to \(40\). JCMCC 74, 95–102 (2010)

    MathSciNet  MATH  Google Scholar 

  17. Dinitz, J.H., Ling, A.: The Hamilton-Waterloo problem: the case of triangle-factors and one Hamilton cycle. J. Combin. Des. 17, 160–176 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Dinitz, J.H., Ling, A.C.H.: The Hamilton-Waterloo problem: The case of triangle-factors and Hamilton cycles: The case \(n\equiv 3~(\text{ mod } \;\;\; 18)\). J. Combin. Math. Combin. Comput. 70, 143–147 (2009)

    MathSciNet  MATH  Google Scholar 

  19. Franek, F., Holub, J., Rosa, A.: Two-factorizations of small complete graphs. II. The case of 13 vertices. J. Combin. Math. Combin. Comput. 51, 89–94 (2004)

    MathSciNet  MATH  Google Scholar 

  20. Franek, F., Rosa, A.: Two-factorizations of small complete graphs. J. Stat. Plann. Inference 86, 435–442 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  21. Horak, P., Nedela, R., Rosa, A.: The Hamilton–Waterloo problem: the case of Hamilton cycles and triangle-factors. Discrete Math. 284, 181–188 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  22. Jordon, H., Morris, J.: Cyclic hamiltonian cycle systems of the complete graph minus a \(1\)-factor. Discrete Math. 308, 2440–2449 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. Keranen, M.S., Ozkan, S.: The Hamilton–Waterloo Problem with 4-cycles and a single factor of \(n\)-cycles. Graphs Combin. 29, 1827–1837 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Liu, J.: The equipartite Oberwolfach problem with uniform tables. J. Combin. Theory Ser. A 101, 20–34 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  25. Liu, J., Lick, D.R.: On \(\lambda \)-fold equipartite Oberwolfach problem with uniform table sizes. Ann. Comb. 7, 315–323 (2003)

    Article  MathSciNet  Google Scholar 

  26. Piotrowski, W.L.: The solution of the bipartite analogue of the Oberwolfach problem. Discrete Math. 97, 339–356 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  27. Rinaldi, G., Traetta, T.: Graph products and new solutions to Oberwolfach problems. Electron. J. Combin. 18, P52 (2011)

    MathSciNet  Google Scholar 

  28. Shalaby, N.: Skolem and Langford sequences. In: Colbourn, C.J., Dinitz, J.H. (eds.) CRC Handbook of Combinatorial Designs, pp. 612–616. CRC Press, Boca Raton (2006)

    Google Scholar 

  29. Traetta, T.: A complete solution to the two-table Oberwolfach problems. J. Combin. Theory Ser. A 120, 984–997 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  30. West, D.: Introduction to Graph Theory. Prentice Hall, New Jersey (1996)

    MATH  Google Scholar 

Download references

Acknowledgments

M. Buratti is supported by MIUR (project “Disegni combinatori, grafi e loro applicazioni”, PRIN 2008). P. Danziger is supported by the NSERC Discovery program. The bulk of this work was carried out when the second author visited University Sapienza of Rome. The support and hospitality of the department during this visit was greatly appreciated.

Author information

Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università di Perugia, via Vanvitelli 1, 06123, Perugia, Italy

    Marco Buratti

  2. Department of Mathematics, Ryerson University, 350 Victoria Street, Toronto, ON, M5B 2K3, Canada

    Peter Danziger

Authors
  1. Marco Buratti
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peter Danziger
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marco Buratti.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Buratti, M., Danziger, P. A Cyclic Solution for an Infinite Class of Hamilton–Waterloo Problems. Graphs and Combinatorics 32, 521–531 (2016). https://doi.org/10.1007/s00373-015-1582-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-015-1582-x

Keywords

Search

Navigation