Skip to main content
Log in

A Note on Cyclic m-cycle Systems of K r(m)

  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

It was proved by Buratti and Del Fra that for each pair of odd integers r and m, there exists a cyclic m-cycle system of the balanced complete r-partite graph K r(m) except for the case when r=m=3. In this note, we study the existence of a cyclic m-cycle system of K r(m) where r or m is even. Combining the work of Buratti and Del Fra, we prove that cyclic m-cycle systems of K r(m) exist if and only if (a) K r(m) is an even graph (b) (r, m)≠ (3, 3) and (c) (r,m)≢ (t , 2) (mod 4) where t ∈ {2,3}.

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alspach, B., Gavlas, H.: Cycle decompositions of K n and K n I, J. Combin. Theory Ser. B 81, 77–99 (2001)

  2. Buratti, M., Del Fra, A.: Existence of cyclic k-cycle systems of the complete graph. Discrete Math. 261, 113–125 (2003)

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

  4. Bryant, D., Gavlas, H., Ling, A.: Skolem-type difference sets for cycle systems. The Electronic Journal of Combinatorics 10, 1–12 (2003)

    Google Scholar 

  5. Fu, H.L., Wu, S.L.: Cyclically decomposing the complete graph into cycles. Discrete Math. 282, 267–273 (2004)

    Google Scholar 

  6. Kotzig, A: Decompositions of a complete graph into 4k-gons. (Russian) Mat. Fyz. Casopis Sloven. Akad. Vied 15, 229–233 (1965)

  7. Peltesohn, R.: Eine Lösung der beiden Heffterschen Differenzenprobleme. Compositio Math. 6, 251–257 (1938)

    Google Scholar 

  8. Rosa, A.: On cyclic decompositions of the complete graph into (4m + 2)-gons. Mat. Fyz. Casopis Sloven. Akad. Vied 16, 349–352 (1966)

    Google Scholar 

  9. Rosa, A.: On cyclic decompositions of the complete graph into polygons with odd number of edges (Slovak). Časopis Pest. Mat. 91, 53–63 (1966)

    Google Scholar 

  10. Rosa, A.: On decompositions of the complete graph into 4k-gons. (Russian) Mat. Fyz. Casopis Sloven. Akad. Vied 17, 242–246 (1967)

    Google Scholar 

  11. Šajna, M.: Cycle decompositions III: Complete graphs and fixed length cycles. J. Combin. Des. 10, 27–78 (2002)

    Google Scholar 

  12. Vietri, A.: Cyclic k-cycle system of order 2km + k; a solution of the last open cases. J. Combin. Des. 12, 299–310 (2004)

    Google Scholar 

  13. Wu, S.L., Fu, H.L.: Cyclic m-cycle systems with m≤ 32 or m=2 q with q a prime power. J. Combin. Des. 14, 66–81 (2006)

  14. Wu, S.L., Fu, H.L.: Maximum cyclic 4-cycle packings of the complete multipartite graph, (in preprint)

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wu, SL., Fu, HL. A Note on Cyclic m-cycle Systems of K r(m) . Graphs and Combinatorics 22, 427–432 (2006). https://doi.org/10.1007/s00373-006-0658-z

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-006-0658-z

Keywords

Navigation