An Application of Completely Separating Systems to Graph Labeling

  • Leanne Rylands
  • Oudone Phanalasy
  • Joe Ryan
  • Mirka Miller
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8288)

Abstract

In this paper a known algorithm used for the construction of completely separating systems (CSSs), Roberts’ Construction, is modified and used in a variety of ways to build CSSs. The main interest is in CSSs with different block sizes. A connection between CSSs and vertex antimagic edge labeled graphs is then exploited to prove that various non-regular graphs are antimagic. An outline for an algorithm which produces some of these non-regular graphs together with a vertex antimagic edge labeling is presented.

Keywords

completely separating system antimagic labeling non-regular graph 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alon, A., Kaplan, G., Lev, A., Rodity, Y., Yuster, R.: Dense graphs are antimagic. J. Graph Theory 47(4), 297–309 (2004)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bodendiek, R., Walther, G.: On number theoretical methods in graph labelings. Res. Exp. Math. 21, 3–25 (1995)MathSciNetGoogle Scholar
  3. 3.
    Chawathe, P.D., Krishna, V.: Antimagic labelings of complete m-ary trees. In: Number Theory and Discrete Mathematics. Trends Math., pp. 77–80. Birkhäuser, Basel (2002)CrossRefGoogle Scholar
  4. 4.
    Dickson, T.J.: On a problem concerning separating systems of a finite set. J. Combinatorial Theory 7, 191–196 (1969)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Gallian, J.A.: A dynamic survey of graph labeling. Electron. J. Combin. 19(\(\sharp\)DS6) (2012)Google Scholar
  6. 6.
    Hartsfield, N., Ringel, G.: Pearls in Graph Theory: A Comprehensive Introduction. Academic Press Inc., Boston (1990)MATHGoogle Scholar
  7. 7.
    Hefetz, D.: Anti-magic graphs via the combinatorial nullstellensatz. J. Graph Theory 50(4), 263–272 (2005)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Kotzig, A., Rosa, A.: Magic valuations of complete graphs. Publ. Centre de Recherches Mathématiques, Université de Montréal 175, CRM–175 (1972)Google Scholar
  9. 9.
    Kotzig, A.: On certain vertex-valuations of finite graphs. Utilitas Math. 4, 261–290 (1973)MathSciNetMATHGoogle Scholar
  10. 10.
    Kotzig, A., Rosa, A.: Magic valuations of finite graphs. Canad. Math. Bull. 13, 451–461 (1970)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    MacDougall, J.A., Miller, M., Slamin, Wallis, W.D.: Vertex magic total labelings of graphs. Utilitas Math. 61, 3–21 (2002)MathSciNetMATHGoogle Scholar
  12. 12.
    MacDougall, J.A., Miller, M., Wallis, W.D.: Vertex magic total labelings of wheels and related graphs. Utilitas Math. 62, 175–183 (2002)MathSciNetMATHGoogle Scholar
  13. 13.
    Phanalasy, O.: Covering Separating Sytems. Master’s thesis, Northern Territory University, Australia (1999)Google Scholar
  14. 14.
    Phanalasy, O., Miller, M., Rylands, L., Lieby, P.: On a relationship between completely separating systems and antimagic labeling of regular graphs. In: Iliopoulos, C.S., Smyth, W.F. (eds.) IWOCA 2010. LNCS, vol. 6460, pp. 238–241. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  15. 15.
    Ramsay, C., Roberts, I.T., Ruskey, F.: Completely separating systems of k-sets. Discrete Math. 183(1–3), 265–275 (1998)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Roberts, I.T.: Extremal Problems and Designs on Finite Sets. Ph.D. thesis, Curtin University of Technology, Australia (1999)Google Scholar
  17. 17.
    Sedláček, J.: Problem 27. In: Theory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963), pp. 163–164. Publ. House Czechoslovak Acad. Sci, Prague (1964)Google Scholar
  18. 18.
    Stewart, B.M.: Magic graphs. Canad. J. Math. 18, 1031–1056 (1966)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Wallis, W.D.: Magic graphs. Birkhäuser Boston Inc., Boston (2001)CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Leanne Rylands
    • 1
  • Oudone Phanalasy
    • 2
    • 3
  • Joe Ryan
    • 4
  • Mirka Miller
    • 2
    • 5
    • 6
    • 7
  1. 1.School of Computing, Engineering and MathematicsUniversity of Western SydneySydneyAustralia
  2. 2.School of Mathematical and Physical SciencesUniversity of NewcastleNewcastleAustralia
  3. 3.Department of MathematicsNational University of LaosVientianeLaos
  4. 4.School of Electrical Engineering and Computer ScienceUniversity of NewcastleNewcastleAustralia
  5. 5.Department of MathematicsUniversity of West BohemiaPilsenCzech Republic
  6. 6.Department of InformaticsKing’s College LondonLondonUK
  7. 7.Department of MathematicsITB BundungBundungIndonesia

Personalised recommendations