Skip to main content
SpringerLink
Log in

Decomposition of Product Graphs into Paths and Cycles of Length Four

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

Abstract

Let \(P_{k}\) and \(C_k\) respectively denote a path and a cycle on \(k\) vertices. In this paper we give necessary and sufficient conditions for the existence of \(\{P_{5}, C_4\}_{\{p, q\}}\)-decomposition of tensor product and cartesian product of complete graphs. Further we extent such decomposition to complete multipartite graphs.

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.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Abueida, A.A., Daven, M.: Multidesigns for graph-pairs of order 4 and 5. Graphs Combin. 19, 433–447 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  2. Abueida, A.A., Daven, M.: Multidecompositions of the complete graph. Ars Combin. 72, 17–22 (2004)

    MATH  MathSciNet  Google Scholar 

  3. Abueida, A.A., Daven, M., Roblee, K.J.: Multidesigns of the \(\lambda \)-fold complete graph-pairs of orders 4 and 5. Australas. J. Combin. 32, 125–136 (2005)

    MATH  MathSciNet  Google Scholar 

  4. Abueida, A.A., O’Neil, T.: Multidecomposition of \(\lambda K_m\) into small cycles and claws. Bull. Inst. Comb. Appl. 49, 32–40 (2007)

    MATH  MathSciNet  Google Scholar 

  5. Abueida, A.A., Daven, M.: Multidecompositions of several graph products. Graphs Combin. 29, 315–326 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  6. Alspah, B., Bermond, J.C., Sotteau, D.: Decomposition into cycles I: Hamilton decompositions. In: Hahn, G., et al. Cycles and Rays. Kluwer Academic Publisher, Dordrecht, pp. 9–18 (1990)

  7. Alspach, B., Gavlas, H.: Cycle decompositions of \(K_n\) and \(K_n-I,\). J. Combin. Theory Ser. B 81, 77–99 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  8. Billington, E.J., Hoffman, D.G., Lindner, C.C., Meszka, M.: Almost resolvable minimum coverings of complete graphs with 4-cycles. Australas. J. Combin. 50, 73–85 (2011)

    MATH  MathSciNet  Google Scholar 

  9. Bondy, J.A., Murty, U.R.S.: Graph Theory with Applications. The Macmillan Press Ltd, New York (1976)

    Book  MATH  Google Scholar 

  10. Chartrand, G., Lesniak, L.: Graphs and Digraphs, 2nd edn. Wadsworth, Belmont (1986)

    MATH  Google Scholar 

  11. Chou, C.C., Fu, C.M., Huang, W.C.: Decomposition of \(K_{m, n}\) into short cycles. Discrete Math. 197(198), 195–203 (1999)

    Article  MathSciNet  Google Scholar 

  12. Chou, C.C., Fu, C.M.: Decomposition of \(K_{m, n}\) into 4-cycles and \(2t\)-cycles. J. Combin. Optim. 14, 205–218 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  13. Dejter, I.J., Lindner, C.C., Meszka, M., Rodger, C.A.: Almost resolvable 4-cycle systems. J. Combin. Math. Combin. Comput. 63, 173–182 (2007)

    MATH  MathSciNet  Google Scholar 

  14. Hoffman, D.G., Pike, D.: 4-Cycle decomposition of the cartesian product of two complete graphs. J. Combin. Math. Combin. Comput. 28, 215–226 (1998)

    MATH  MathSciNet  Google Scholar 

  15. Jeevadoss, S., Muthusamy, A.: Sufficient condition for \( \{C_4, \, C_{2t}\} \) - decomposition of \(K_{2m,2n}\)—an improved bound. In: Arumugam, S., Smyth, B. (eds.) Combinatorial Algorithms, IWOCA 2012, LNCS, vol. 7643. Springer, Berlin, pp. 143–147 (2012)

  16. Jeevadoss, S., Muthusamy, A.: Decomposition of complete bipartite graphs into paths and cycles. Discrete Math. 331, 98–108 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  17. Lee, H.-C., Chu, Y.-P.: Multidecompositions of the balanced complete bipartite graphs into paths and stars. ISRN Combin. (2013). doi:10.1155/2013/398473

  18. Lee, H.-C.: Multidecompositions of complete bipartite graphs into cycles and stars. Ars Combin. 108, 355–364 (2013)

    MATH  MathSciNet  Google Scholar 

  19. Lindner, C.C., Rodger, C.A.: Design theory, 2nd edn. CRC Press, Boca Raton (2009)

    MATH  Google Scholar 

  20. Shyu, T.-W.: Decompositions of complete graphs into paths and cycles. Ars Combin. 97, 257–270 (2010)

    MATH  MathSciNet  Google Scholar 

  21. Shyu, T.-W.: Decompositions of complete graphs into paths and stars. Discrete Math. 330, 2164–2169 (2010)

    Article  MathSciNet  Google Scholar 

  22. Shyu, T.-W.: Decomposition of complete graphs into cycles and stars. Graphs Combin. 29, 301–313 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  23. Shyu, T.-W.: Decomposition of complete bipartite graphs into paths and stars with same number of edges. Discrete Math. 313, 865–871 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  24. Sotteau, D.: Decomposition of \(K_{m, n}\) \((K_{m, n}^{*})\) into cycles (circuits) of length \(2k\). J. Combin. Theory Ser. B 30, 75–81 (1981)

  25. Tarsi, M.: Decomposition of complete multigraph into simple paths: nonbalanced Handcuffed designs. J. Combin. Theory Ser. A 34, 60–70 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  26. Truszczyński, M.: Note on the decomposition of \(\lambda K_{m, n}(\lambda K^{*}_{m, n})\) into paths. Discrete Math. 55, 89–96 (1985)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors are grateful to the anonymous referees for their valuable suggestions and comments, which improved the presentation of the paper. The first author thanks the University Grants Commission for its support (Grant No.F.4-7/2008(BSR)/11-105/2008(BSR)/ December 2012). The second author thank the Department of Science and Technology, Government of India, New Delhi for its financial support through the Grant No. DST/ SR/ S4/ MS:282/13.

Author information

Authors and Affiliations

  1. Department of Mathematics, Periyar University, Salem, Tamil Nadu, India

    S. Jeevadoss & A. Muthusamy

Authors
  1. S. Jeevadoss
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Muthusamy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Muthusamy.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jeevadoss, S., Muthusamy, A. Decomposition of Product Graphs into Paths and Cycles of Length Four. Graphs and Combinatorics 32, 199–223 (2016). https://doi.org/10.1007/s00373-015-1564-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-015-1564-z

Keywords

Search

Navigation