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Computer Science Today pp 172–183Cite as

Edge-coloring algorithms

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1000))

Abstract

The edge-coloring problem is one of the fundamental problems on graphs, which often appears in various scheduling problems like the file transfer problem on computer networks. In this paper, we survey recent advances and results on the classical edge-coloring problem as well as the generalized edge-coloring problems, called the f-coloring and Φ-coloring problems. In particular we review various upper bounds on the minimum number of colors required to edge-color graphs, and present efficient algorithms to edge-color graphs with a number of colors not exceeding the upper bounds.

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References

  1. Aho, A. V., Hopcroft, J. E., Ullman, J. D.: The Design and Analysis of Computer Algorithms. Addison-Wesley, Reading, MA (1974).

    Google Scholar 

  2. Andersen, L.: On edge colorings of graphs. Math. Scand. 40 (1977) 161–175.

    Google Scholar 

  3. Arnborg, S., Proskurowski, A.: Linear time algorithms for NP-hard problems restricted to partial k-trees. Discrete Appl. Math. 23 (1989) 11–24.

    Article  Google Scholar 

  4. Bodlaender, H. L.: Polynomial algorithms for graph isomorphism and chromatic index on partial k-trees. Journal of Algorithms 11 (1990) 631–643.

    Article  Google Scholar 

  5. Coffman, Jr, E. G., Garey, M. R., Johnson, D. S., LaPaugh, A. S.: Scheduling file transfers. SIAM J. Comput. 14 (1985) 744–780.

    Article  Google Scholar 

  6. Chrobak, M., Nishizeki, T.: Improved edge-coloring algorithms for planar graphs. Journal of Algorithms 11 (1990) 102–116.

    Article  Google Scholar 

  7. Chrobak, M., Yung, M.: Fast algorithms for edge-coloring planar graphs. Journal of Algorithms 10 (1989) 35–51.

    Article  Google Scholar 

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

    Article  Google Scholar 

  9. Ehrenfeucht, A., Faber, V., Kierstead, H. A.: A new method of proving theorems on chromatic index. Discrete Mathematics 52 (1984) 159–164.

    Article  Google Scholar 

  10. Fiorini, S., Wilson, R. J.: Edge-Colouring of Graphs, Pitman, London (1977).

    Google Scholar 

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

    Article  Google Scholar 

  12. Gabow, H. N., Nishizeki, T., Kariv, O., Leven, D., Terada, O.: Algorithms for edgecoloring graphs. Tech. Rep. TRECIS-8501, Tohoku Univ. (1985).

    Google Scholar 

  13. Garey, M. R., Johnson, D. S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., San Francisco (1979).

    Google Scholar 

  14. Goldberg, M. K.: Edge-colorings of multigraphs: recoloring techniques. Journal of Graph Theory 8 (1984) 122–136.

    Google Scholar 

  15. Hakimi, S. L., Kariv, O.: On a generalization of edge-coloring in graphs. Journal of Graph Theory 10 (1986) 139–154.

    Google Scholar 

  16. Hilton, A. J. W., de Werra, D.: Sufficient conditions for balanced and for equitable edge-colorings of graphs. O. R. Working paper 82/3, Dépt. of Math., école Polytechnique Fédérate de Lausanne, Switzerland (1982).

    Google Scholar 

  17. Hochbaum, D. S., Nishizeki, T., Shmoys, D. B.: A better than “best possible” algorithm to edge color multigraphs. Journal of Algorithms 7 (1986) 79–104.

    Article  Google Scholar 

  18. Holyer, I. J.: The NP-completeness of edge colourings. SIAM J. Comput. 10 (1980) 718–720.

    Article  Google Scholar 

  19. Karloff, H. J., Shmoys, D. B.: Efficient parallel algorithms for edge-coloring problems. Journal of Algorithms 8 (1987) 39–52.

    Article  Google Scholar 

  20. König, D., über Graphen und iher Anwendung auf Determinantentheorie und Mengenlehre. Math. Ann. 77 (1916) 453–465.

    Article  Google Scholar 

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

    Google Scholar 

  22. Marcotte, O.: Exact edge-coloring of graphs without prescribed minors. DIMACS Series in Discrete Mathematics and Theoretical Computer Science 1 (1990) 235–249.

    Google Scholar 

  23. Nakano, S., Nishizeki, T., Saito, N.: On the f-coloring of multigraphs. IEEE Trans. Circuits and Syst. CAS-35 (1988) 345–353.

    Article  Google Scholar 

  24. Nakano, S., Nishizeki, T., Saito, N.: On the fg-coloring of graphs. Combinatorica 10 (1990) 67–80.

    Article  Google Scholar 

  25. Nakano, S., Nishizeki, T.: Scheduling file transfers under port and channel constraints. International Journal of Foundation of Computer Science 4 (1993) 101–115.

    Article  Google Scholar 

  26. Nakano, S., Nishizeki, T.: Approximation algorithms for the f-edge-coloring of multigraphs. Trans. of Japan SIAM 3 (1993) 279–307 (in Japanese).

    Google Scholar 

  27. Nishizeki, T., Kashiwagi, K.: On the 1.1 edge-coloring of multigraphs. SIAM J. Disc. Math. 3 (1990) 391–410.

    Article  Google Scholar 

  28. Seymour, P. D.: On multicolouring of cubic graphs, and conjecture of Fulkerson and Tutte. Proc. London Math. Soc. 38 (1979) 423–460.

    Google Scholar 

  29. Seymour, P. D.: Colouring series-parallel graphs. Combinatorica 10 (1990) 379–392.

    Article  Google Scholar 

  30. Shannon, C. E.: A theorem on coloring the lines of a network. J. Math. Phys. 28 (1949) 148–151.

    Google Scholar 

  31. Terada, O., Nishizeki, T.: Approximate algorithms for the edge-coloring of graphs. Trans. IEICE Japan J65-D (1982) 1382–1389 (in Japanese).

    Google Scholar 

  32. Vizing, 2. G.: On an estimate of the chromatic class of a p-graph. Discret Analiz 3 (1964) 25–30 (in Russian).

    Google Scholar 

  33. Vizing, V. G.: The chromatic class of a multigraph. Kibernetica(Kief) 3 (1965) 29–39.

    Google Scholar 

  34. de Werra, D.: Some results in chromatic scheduling. Zeitschrift fur Oper. Res. 18 (1974) 167–175.

    Article  Google Scholar 

  35. de Werra, D.: A few remarks on chromatic scheduling. in Combinatorial Programming: Methods and Applications, ed. Roy, B., D. Reidel Pub. Comp., Dordrecht-Holland (1975) 337–342.

    Google Scholar 

  36. Zhou, X., Nakano, S., Nishizeki, T.: A linear algorithm for edge-coloring partial k-tree. Th. Lengauer (Ed.), Algorithms — ESA'93, Proc. First Annual European Symposium, Lect. Notes in Comp. Sci., Vol. 726, Springer-Verlag (1993) 409–418.

    Google Scholar 

  37. Zhou, X., Nakano, S., Nishizeki, T.: A parallel algorithm for edge-coloring partial k-tree. in: E.M. Schmidt, S. Skyum (Eds.), Algorithm Theory — SWAT '94, Proc. 4th Scand. Workshop on Algorithm Theory, Lect. Notes in Comp. Sci., Vol. 824, Springer-Verlag (1994) 359–369.

    Google Scholar 

  38. Zhou, X., Nakano, S., Suzuki, H., Nishizeki, T.: An efficient algorithm for edgecoloring series-parallel multigraphs. in: I. Simon (Ed.), LATIN'92, Proc. 1st Latin American Symposium on Theoretical Informatics, Lect. Notes in Comp. Sci., Vol. 583, Springer-Verlag (1992) 516–529.

    Google Scholar 

  39. Zhou, X., Nishizeki, T.: Edge-coloring and f-coloring for various classes of graphs. in: D-Z. Du, X-S. Zhang (Eds.), Algorithms and Computation, Proc. 5th In. Symposium (ISAAC'94), Lect. Notes in Comp. Sci., Vol. 834, Springer-Verlag (1994) 199–207.

    Google Scholar 

  40. Zhou, X., Nishizeki, T.: Optimal parallel algorithms for edge-coloring partial k-trees with bounded degrees. Trans. IEICE Japan E78-A (1995) 463–469.

    Google Scholar 

  41. Zhou, X., Nishizeki, T.: Simple reductions of f-coloring to edge-colorings. Proc. of First Ann. Int. Comput. and Combinatorics Conf., Lect. Notes in Comp. Sci., Springer-Verlag, to appear.

    Google Scholar 

  42. Zhou. X., Suzuki, H., Nishizeki, T.: A linear algorithm for edge-coloring seriesparallel multigraphs. Journal of Algorithms, to appear.

    Google Scholar 

  43. Zhou, X., Suzuki, H., Nishizeki, T.: Sequential and parallel algorithms for edgecoloring series-parallel multigraphs. Proc. of the Third Conference on Integer Programming and Combinatorial Optimization (1993) 129–146.

    Google Scholar 

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Authors and Affiliations

  1. Graduate School of Information Sciences, Tohoku University, 980-77, Sendai, Japan

    Shin-ichi Nakano, Xiao Zhou & Takao Nishizeki

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  1. Shin-ichi Nakano
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  2. Xiao Zhou
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  3. Takao Nishizeki
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Jan van Leeuwen

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© 1995 Springer-Verlag Berlin Heidelberg

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Nakano, Si., Zhou, X., Nishizeki, T. (1995). Edge-coloring algorithms. In: van Leeuwen, J. (eds) Computer Science Today. Lecture Notes in Computer Science, vol 1000. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0015243

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  • DOI: https://doi.org/10.1007/BFb0015243

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-60105-0

  • Online ISBN: 978-3-540-49435-5

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