Advertisement

More Lower Bounds for Weak Sense of Direction: The Case of Regular Graphs

  • Paolo Boldi
  • Sebastiano Vigna‡
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1914)

Abstract

A graph G with n vertices and maximum degree δG cannot be given weak sense of direction using less than δG colours. It is known that n colours are always sufficient, and it was conjectured that just δG + 1 are really needed, that is, one more colour is sufficient. Nonetheless, it has just been shown [2] that for sufficiently large n there are graphs requiring Ω(n/log n) more colours than δG. In this paper, using recent results in asymptotic graph enumeration, we show not only that (somehow surprisingly) the same bound holds for regular graphs, but also that it can be improved to Ω(n log log n/ log n). We also show that Ω (dG√log log dG) colours are necessary, where dG is the degree of G.

Keywords

Random Graph Regular Graph Weak Sense Additional Colour Local Naming 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Paolo Boldi and Sebastiano Vigna. Complexity of deciding sense of direction. SIAM J. Compute 29(3):779–789, 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Paolo Boldi and Sebastiano Vigna. Lower bounds for weak sense of direction. In Structure, Information and Communication Complexity. Proc. 7th Colloquium SIROCCO 2000, Proceedings In Informatics. Carleton Scientific, 2000.Google Scholar
  3. 3.
    Bĺa BollobŚ. Random Graphs. Academic Press, London, 1985.Google Scholar
  4. 4.
    Paola Flocchini, Bernard Mans, and Nicola Santoro. On the impact of sense of direction on message complexity. Inform. Process. Lett, 63(1):23–31, 1997.CrossRefMathSciNetGoogle Scholar
  5. 5.
    Paola Flocchini, Bernard Mans, and Nicola Santoro. Sense of direction: Definitions, properties, and classes. Networks, 32(3):165–180, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Paola Flocchini, Bernard Mans, and Nicola Santoro. Sense of direction in distributed computing. In Proc. DISC ’98, volume 1499 of Lecture Notes in Computer Science, pages 1–15, 1998.Google Scholar
  7. 7.
    Ronald L. Graham, Donald E. Knuth, and Oren Patashnik. Concrete Mathematics. Addison-Wesley, second edition, 1994.Google Scholar
  8. 8.
    Brendan D. McKay and Nicholas C. Wormald. Asymptotic enumeration by degree sequence of graphs of high degree. European J. Combin., ll(6):565–580, 1990.MathSciNetGoogle Scholar
  9. 9.
    Nicholas C. Wormald. Personal electronic communication.Google Scholar
  10. 10.
    Nicholas C. Wormald. Models of random regular graphs. In Surveys in combinatorics, 1999 (Canterbury), pages 239–298. Cambridge Univ. Press, Cambridge, 1999.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Paolo Boldi
    • 1
  • Sebastiano Vigna‡
    • 1
  1. 1.Dipartimento di Scienze deH’InformazioneUniversità degli Studi di MilanoItaly

Personalised recommendations