On \(\sigma \)-Tripartite Labelings of Odd Prisms and Even Möbius Ladders

  • Wannasiri WannasitEmail author
  • Saad El-Zanati


A common question in the study of graph decompositions is when does a graph G decompose the complete graph or the complete graph with a 1-factor removed or added. It is known that a \(\sigma \)-tripartite labeling of a tripartite graph G with n edges can be used to obtain a cyclic G-decomposition of \(K_{2nt+1}\) for every positive integer t. Moreover, it can be used to obtain a cyclic G-decomposition of both \(K_{2nt+2}-I\) and \(K_{2nt}+I\), where I is a 1-factor. We show that if G is an odd prism on 10 or more vertices or an even Möbius ladder, then G admits a \(\sigma \)-tripartite labeling.


Cyclic G-designs Cubic Tripartite graphs \(\sigma \)-tripartite labelings 

Mathematics Subject Classification




The authors wish to thank an anonymous referee for several helpful suggestions that improved the presentation of the results in this paper. This research was supported by the Thailand Research Fund (TRF) and Chiang Mai University, Grant No. TRG5880080.


  1. 1.
    Adams, P., Bryant, D., Buchanan, M.: A survey on the existence of G-designs. J. Combin. Des. 16, 373–410 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bryant, D., El-Zanati, S.: Graph decompositions. In: Colbourn, C.J., Dinitz, J.H. (eds.) Handbook of Combinatorial Designs, 2nd edn, pp. 477–485. Chapman, Boca Raton (2007)Google Scholar
  3. 3.
    Bunge, R.C., Chantasartrassmee, A., El-Zanati, S.I., Eynden, C.Vanden: On cyclic decompositions of complete graphs into tripartite graphs. J. Gr. Theory. 72, 90–111 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    El-Zanati, S.I., Eynden, C.Vanden: On Rosa-type labelings and cyclic graph decompositions. Math. Slov. 59, 1–18 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    El-Zanati, S.I., Eynden, C.Vanden, Punnim, N.: On the cyclic decomposition of complete graphs into bipartite graphs. Australas. J. Combin. 24, 209–219 (2001)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Frucht, R., Gallian, J.: Labeling prisms. Ars Combin. 26, 69–82 (1988)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Gallian, J.A.: Labeling prisms and prism related graphs. Congr. Numer. 59, 89–100 (1987)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Gallian, J.A.: A dynamic survey of graph labeling. Electron. J. Combin., Dynamic Survey DS6 (2015).
  9. 9.
    Pasotti, A.: Constructions for cyclic Moebius ladder systems. Discret. Math. 310, 3080–3087 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Read, R.C., Wilson, R.J.: An Atlas of Graphs. Clarendon Press, Oxford (1998)zbMATHGoogle Scholar
  11. 11.
    Rosa, A.: On certain valuations of the vertices of a graph. In: Théorie des graphes, journées internationales d’études, Rome, Dunod, Paris, 1967) 349–355 (1966)Google Scholar
  12. 12.
    Šajna, M.: Mateja Decomposition of the complete graph plus a 1-factor into cycles of equal length. J. Combin. Des. 11, 170–207 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Vietri, A.: Graceful labellings for an infinite class of generalised Petersen graphs. Ars Combin. 81, 247–255 (2006)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Vietri, A.: A new infinite family of graceful generalised Petersen graphs, via “graceful collages” again. Australas. J. Combin. 41, 273–282 (2008)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Wannasit, W., El-Zanati, S.I.: On graceful cubic graphs. Congr. Numer. 208, 167–182 (2011)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Wannasit, W., El-Zanati, S.: On cyclic \(G\)-designs where \(G\) is a cubic tripartite graph. Discret. Math. 312, 293–305 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Wannasit, W., El-Zanati, S.I.: On free \(\alpha \)-labelings of cubic bipartite graphs. J. Combin. Math. Combin. Computing 82, 269–293 (2012)MathSciNetzbMATHGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2017

Authors and Affiliations

  1. 1.Department of MathematicsChiang Mai UniversityChiang MaiThailand
  2. 2.Department of MathematicsIllinois State UniversityNormalUSA

Personalised recommendations