An Algorithmic Embedding of Graphs via Perfect Matchings

  • Vojtech Rödl
  • Andrzej Ruciński
  • Michelle Wagner
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1518)

Abstract

Recently Komlós, Sárközy, and Szemerédi proved a striking result called the blow-up lemma that, loosely speaking, enables one to embed any bounded degree graph H as a spanning subgraph of an e-regular graph G. The first proof given by Komlós, Sárközy, and Szemerédi was based on a probabilistic argument [8]. Subsequently, they derandomized their approach to provide an algorithmic embedding in [9]. In this paper we give a different proof of the algorithmic version of the blow-up lemma. Our approach is based on a derandomization of a probabilistic proof of the blow-up lemma given in [13]. The derandomization utilizes the Erdös-Selfridge method of conditional probabilities and the technique of pessimistic estimators.

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References

  1. 1.
    N. Alon, R. Duke, H. Leffman, V. Rödl, and R. Yuster, “The algorithmic aspects of the regularity lemma”, Journal of Algorithms, vol. 16 (1994), pp. 80–109.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    N. Alon, V. Rödl, and A. Ruciński, “Perfect matchings in ε-regular graphs”, The Electronic J. of Combin., vol. 5(1) (1998), # R13.Google Scholar
  3. 3.
    N. Alon and J. Spencer, The Probabilistic Method, Wiley, New York, 1992.MATHGoogle Scholar
  4. 4.
    P. Erdös and J. L. Selfridge, “On a combinatorial game”, Journal of Combinatorial Theory, Series A, vol. 14 (1973), pp. 298–301.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    A. Hajnal and E. Szemerédi, “Proof of a conjecture of Erdös”, Combinatorial Theory and its Applications, vol. II (P. Erdós, A. Rényi, and V.T. Sós eds), Colloq. Math. Soc. J. Bolyai 4, North Holland, Amsterdam, 1970, pp. 601–623.Google Scholar
  6. 6.
    J. E. Hopcroft R. M. Karp, “An n5/2 algorithm for maximum matchings in bipartite graphs”, SIAM J. Comput, vol. 2 (1973), pp. 225–231.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    S. Janson, T. łuczak, and A. Ruciński, Topics in Random Graphs, Wiley, New York, 1999.Google Scholar
  8. 8.
    J. Komlós, G. N. Sárközy, and E. Szemerédi, “Blow-up lemma”, Combinatorica, vol. 17 (1997), pp. 109–123.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    J. Komlós, G. N. Sárközy, and E. Szemerédi, “An algorithmic version of the blowup lemma”, Random Structures and Algorithms, vol. 12 (1998), pp. 297–312.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    J. Komlós, G. N. Sárközy and E. Szemerédi “On the Pósa-Seymour conjecture J. Graph Theory”, Journal of Graph Theory, to appearGoogle Scholar
  11. 11.
    J. Komlós, G. N. Sárközy and E. Szemerédi “Proof of the Alon-Yuster conjecture” submittedGoogle Scholar
  12. 12.
    P. Raghavan, “Probabilistic construction of deterministic algorithms: Approximating packing integer programs”, Journal of Computer and System Sciences, vol. 37 (1988), pp. 130–143.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    V. Rödl and A. Ruciński, “Perfect matchings in ε-regular graphs and the blow-up lemma”, submitted.Google Scholar
  14. 14.
    N. Sauer and J. Spencer, “Edge disjoint placement of graphs”, J. Comb. Th. B, vol. 25 (1978), pp. 295–302.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    E. Szemerédi “Partitions of graphs” Problems Combin. et Theorie des graphes, Edition du C.N.R.S. vol. 260 (1978), pp. 399–402.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Vojtech Rödl
    • 1
  • Andrzej Ruciński
    • 1
  • Michelle Wagner
    • 1
  1. 1.Department of Mathematics and Computer ScienceEmory UniversityAtlantaUSA

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