Abstract
Given 2-factors \(R\) and \(S\) of order \(n\), let \(r\) and \(s\) be nonnegative integers with \(r+s=\lfloor \frac{n-1}{2}\rfloor \), the Hamilton–Waterloo problem asks for a 2-factorization of \(K_n\) if \(n\) is odd, or of \(K_n-I\) if \(n\) is even, in which \(r\) of its 2-factors are isomorphic to \(R\) and the other \(s\) 2-factors are isomorphic to \(S\). In this paper, we solve the problem for the case of triangle-factors and heptagon-factors for odd \(n\) with 3 possible exceptions when \(n=21\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adams, P., Billington, E.J., Bryant, D.E., El-Zanati, S.I.: On the Hamilton–Waterloo problem. Graphs Comb. 18, 31–51 (2002)
Alspach, B., Häggkvist, R.: Some observations on the Oberwolfach problem. J. Graph Theory 9, 177–187 (1985)
Alspach, B., Schellenberg, P.J., Stinson, D.R., Wagner, D.: The Oberwolfach problem and factors of uniform odd length cycles. J. Comb. Theory Ser. A 52, 20–43 (1989)
Bryant, D.E., Danziger, P.: On bipartite 2-factorizations of \(k_n-I\) and the Oberwolfach problem. J. Graph Theory 68, 22–37 (2011)
Bryant, D.E., Danziger, P., Dean, M.: On the Hamilton–Waterloo problem for bipartite 2-factors. J. Comb. Des. 21, 60–80 (2013)
Danziger, P., Quattrocchi, G., Stevens, B.: The Hamilton–Waterloo problem for cycle sizes 3 and 4. J. Comb. Des. 17, 342–352 (2009)
Dinitz, J.H., Ling, A.C.H.: The Hamilton–Waterloo problem with triangle-factors and Hamilton cycles: the case \(n\equiv 3 ({\rm mod\,18})\). J. Comb. Math. Comb. Comput. 70, 143–147 (2009)
Dinitz, J.H., Ling, A.C.H.: The Hamilton–Waterloo problem: the case of triangle-factors and one Hamilton cycle. J. Comb. Des. 17, 160–176 (2009)
Fu, H.L., Huang, K.C.: The Hamilton–Waterloo problem for two even cycles factors. Taiwan. J. of Math. 12, 933–940 (2008)
Hoffman, D., Schellenberg, P.: The existence of \(C_k\)-factorizations of \(K_{2n}-F\). Discrete Math. 97, 243–250 (1991)
Horak, P., Nedela, R., Rosa, A.: The Hamilton–Waterloo problem: the case of Hamilton cycles and triangle-factors. Discrete Math. 284, 181–188 (2004)
Lei, H., Fu, H.L., Shen, H.: The Hamilton–Waterloo problem for Hamilton cycles and \(C_{4k}\)-factors. Ars Comb. 100, 341–347 (2011)
Lei, H., Shen, H.: The Hamilton–Waterloo problem for Hamilton cycles and triangle-factors. J. Comb. Des. 20, 305–316 (2012)
Liu, J.: The equipartite Oberwolfach problem with uniform tables. J. Comb. Theory, Ser. A 101, 20–34 (2003)
Acknowledgments
The authors would like to express their deep gratefulness to the reviewers for their detail comments and valuable suggestions. The work of Hung-Lin Fu was partially supported by NSC 100-2115-M-009-005-MY3.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lei, H., Fu, HL. The Hamilton–Waterloo Problem for Triangle-Factors and Heptagon-Factors. Graphs and Combinatorics 32, 271–278 (2016). https://doi.org/10.1007/s00373-015-1570-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00373-015-1570-1