Skip to main content
Springer Nature Link
Log in

The complete grapht-packings andt-coverings

  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

The aim of this paper is to introduce the notions of floor and ceiling functions related to graph edge-decompositions intot mutually isomorphic parts. It is shown that, given any natural numbert, uniform and extremal t-packing andt-covering exist for each complete graph and each complete bipartite graph. Extremal in this context means that both a remainder and a surplus are absolutely minimum. In proofs, decompositions of multigraphs into matchings are involved. Open problems and conjectures are stated.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. Bosák, J.: Decompositions of Graphs. [Slovak:] Bratislava: Veda 1986. [English:] Dordrecht: Kluwer, and Bratislava: Veda 1990

  2. Clapham, C.R.J.: Graphs self-complementary inK n —e. Discrete Math.81, 229–235 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  3. Das, P.K.: Almost selfcomplementary graphs I. Ars Comb.31, 267–276 (1991)

    MATH  Google Scholar 

  4. Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics. Reading, MA, et al." Addison-Wesley 1988

  5. Guidotti, L.: Sulla divisibilita dei graft completi. Riv. Mat. Univ. Parma1, 231–237 (1972)

    MathSciNet  Google Scholar 

  6. Harary, F., Robinson, R.W., Wormald, N.C.: Isomorphic factorisations. I: Complete graphs. Trans. Amer. Math. Soc.242, 243–260 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  7. Harary, F., Robinson, R.W., Wormald, N.C.: Isomorphic factorisations III: Complete multipartite graphs. In (Holton, D., Seberry, J. eds.): Combinatorial Mathematics (Proc. Conf. Canberra 1977). Lect. Notes Math.686, pp. 47–54. Berlin Heidelberg: Springer, and Canberra: Austral. Acad. Sci. 1978

    Google Scholar 

  8. Kirkman, T.P.: On a problem in combinations. Combridge and Dublin Math. J.2, 191–204 (1847)

    Google Scholar 

  9. Quinn, S.J.: Isomorphic factorisations of complete equipartite graphs. J. Graph Theory7, 285–310.(1983)

    Article  MATH  MathSciNet  Google Scholar 

  10. Reiss, M.: Über eine Steinersche combinatorische Aufgabe. J. Reine Angew. Math.56, 326–344 (1859)

    MATH  Google Scholar 

  11. Ringel, G.: Selbstkomplementäre Graphen. Arch. Math.14, 354–358 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  12. Sachs, H.: Über selbstkomplementäire Graphen. Publ. Math. Debrecen9, 270–288 (1962)

    MathSciNet  Google Scholar 

  13. Schönheim, J., Bialostocki, A.: Decomposition ofK n into subgraphs of prescribed type. Arch. Math.31, 105–112 (1978)

    Article  MATH  Google Scholar 

  14. Tarsi, M.: Decomposition of a complete multigraph into simple paths: nonbalanced handcuffed designs. J. Comb. Theory Ser. A34, 60–70 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  15. Wang Jianfang: Isomorphic factorisations of complete equipartite graphs — the proof of the Harary-Robinson-Wormald conjecture. Sci. Sin. (A)25, 1152–1164 (1982)

    MATH  Google Scholar 

  16. Wormald, N.C.: Isomorphic factorizations VII. Regular graphs and tournaments. J. Graph Theory8, 117–122 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  17. Woźniak, M.: Embedding graphs of small size. Proc. Enschede 1991 Conf., (to be published)

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics AGH, Mickiewicza 30, 30-059, Kraków, Poland

    Zdzislaw Skupień

Authors
  1. Zdzislaw Skupień
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Skupień, Z. The complete grapht-packings andt-coverings. Graphs and Combinatorics 9, 353–363 (1993). https://doi.org/10.1007/BF02988322

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02988322

Keywords