Skip to main content
SpringerLink
Log in

Fast parallel and sequential algorithms for edge-coloring planar graphs

extended abstract

  • Parallel Set And Graph Algorithms
  • Conference paper
  • First Online:
VLSI Algorithms and Architectures (AWOC 1988)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 319))

Included in the following conference series:

Abstract

Vizing [21,22,8] proved that any graph G of maximal degree Δ(G) is either Δ or Δ + 1 colorable. For edge-coloring of planar graphs, he showed that all graphs G with Δ(G) ≥ 8 are Δ-colorable. Based on the proof of Vizing's theorem Gabow et al. [10] designed an O(n 2) algorithm for Δ-coloring planar graphs when Δ(G) ≥ 8. Boyar and Karloff [2] proved that this problem belongs to NC if Δ(G) ≥ 23. Their algorithm runs in time O(log3 n) and uses O(n 2) processors on a CRCW PRAM.

We present the following algorithms for Δ-edge coloring planar graphs:

  • A linear time sequential algorithm for graphs G with Δ(G) ≥ 19.

  • An efficient parallel EREW PRAM algorithm for graphs G with Δ ≥ 19. This algorithm runs in time O(log2 n) and uses O(n) processors.

  • An O(n log n) time sequential algorithm for graphs G with Δ(G) ≥ 9.

  • An efficient parallel EREW PRAM algorithm for graphs with Δ ≥ 9. This algorithm runs in time O(log3 n) and uses O(n) processors.

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

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. K.I. Appel, W. Haken, Every planar map is four colorable, Bull. Amer. Math. Soc., vol 82, pp 711–712, 1976.

    Article  MATH  MathSciNet  Google Scholar 

  2. J. Boyar, H. Karloff, Coloring planar graphs in parallel, J. Algorithms 8 (1987) 470–479.

    Article  MATH  MathSciNet  Google Scholar 

  3. N. Chiba, T. Nishizeki, N. Saito, A linear algorithm for five-coloring a planar graph, J. Algorithms 2 (1981) 317–327.

    Article  MATH  MathSciNet  Google Scholar 

  4. R. Cole, J. Hopcroft, On edge coloring bipartite graphs, SIAM J. Comput 11 (1982) 540–546.

    Article  MATH  MathSciNet  Google Scholar 

  5. R.Cole, U.Vishkin, Determistic coin tossing and accelerating cascades: micro and macro techniques for designing parallel algorithms, 18th ACM STOC (1986) 206–219.

    Google Scholar 

  6. R.Cole, U.Vishkin, Approximate and exact parallel scheduling with applications to list, Tree and Graph Problems, 27th IEEE FOCS (1986) 478–491.

    Google Scholar 

  7. G.N. Frederickson, On linear-time algorithms for five-coloring planar graphs, Inform. Proc. Letters 19 (1984) 219–224.

    Article  MATH  MathSciNet  Google Scholar 

  8. S. Fiorini, R.J. Wilson, Edge-Colourings of Graphs, Pitman, London, 1977.

    MATH  Google Scholar 

  9. H.N. Gabow, O. Kariv, Algorithms for edge coloring bipartite graphs and multigraphs, SIAM J. Comput. 11 (1982) 117–129.

    Article  MATH  MathSciNet  Google Scholar 

  10. H.N.Gabow, T.Nishizeki, O.Kariv, D.Leven, O.Terada, Algorithms for edge-coloring graphs, TR-41/85, Department of Computer Science, Tel Aviv University, 1985.

    Google Scholar 

  11. A.V.Goldberg, S.A.Plotkin, G.E.Shannon, Parallel symmetry breaking in sparse graphs, 19th ACM STOC, New York, 1987.

    Google Scholar 

  12. M.Goldberg, T.Spencer, A new parallel algorithm for the maximal independent set problem, 19th IEEE FOCS, 1987.

    Google Scholar 

  13. T.Hagerup, M.Chrobak, K.Diks, Optimal parallel 5-colouring of planar graphs, 14th ICALP, 1987.

    Google Scholar 

  14. I.J. Holyer, The NP-completeness of edge coloring, SIAM J. Comput 10 (1981) 718–720.

    Article  MATH  MathSciNet  Google Scholar 

  15. H. Karloff, D. Shmoys, Efficient parallel algorithms for edge coloring problems, J. Algorithms 8 (1987) 39–52.

    Article  MATH  MathSciNet  Google Scholar 

  16. R.M.Karp, A.Widgerson, A fast parallel algorithm for the maximal independent set problem, 16th ACM STOC, 1987.

    Google Scholar 

  17. M.Luby, A simple parallel algorithm for the maximal independent set problem, 17th ACM STOC, 1985.

    Google Scholar 

  18. D. Leven, Z. Galil, NP-completeness of finding the chromatic index of regular graphs, J. Algorithms 4 (1983) 35–44.

    Article  MATH  MathSciNet  Google Scholar 

  19. G.F. Lev, N. Pippenger, L.G. Valiant, A fast parallel algorithm for routing in permutation networks, IEEE Trans. on Comput. C-30 (1981) 93–100.

    MathSciNet  Google Scholar 

  20. D.Matula, Y.Shiloah, R.Tarjan, Two linear-time algorithms for five-coloring a planar graph, Tech. Rep. No. STAN-CS-80-830, Dep. Comp. Sci., Stanford University.

    Google Scholar 

  21. V.G. Vizing, On the estimate of the chromatic class of a p-graph, Diskret. Analiz 3 (1964) 25–30.

    MathSciNet  Google Scholar 

  22. V.G. Vizing, Critical graphs with a given chromatic number, Diskret. Analiz 5 (1965) 9–17.

    MATH  MathSciNet  Google Scholar 

  23. M.H. Williams, A linear algorithm for coloring planar graphs with five colors, Comput. J. 28,1 (1985) 78–81.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept. of Mathematics, University of California, 92521, Riverside, CA

    Marek Chrobak

  2. IBM Research Division Almaden Research Center, 95120, San Jose, CA

    Moti Yung

Authors
  1. Marek Chrobak
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Moti Yung
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

John H. Reif

Rights and permissions

Reprints and permissions

Copyright information

© 1988 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Chrobak, M., Yung, M. (1988). Fast parallel and sequential algorithms for edge-coloring planar graphs. In: Reif, J.H. (eds) VLSI Algorithms and Architectures. AWOC 1988. Lecture Notes in Computer Science, vol 319. Springer, New York, NY. https://doi.org/10.1007/BFb0040369

Download citation

  • DOI: https://doi.org/10.1007/BFb0040369

  • Published:

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-0-387-96818-6

  • Online ISBN: 978-0-387-34770-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation