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Maximum Induced Matchings of Random Regular Graphs

  • Hilda Assiyatun
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3330)

Abstract

An induced matching of a graph G = (V,E) is a matching \({\mathcal M}\) such that no two edges of \({\mathcal M}\) are joined by an edge of E/\({\mathcal M}\) In general, the problem of finding a maximum induced matching of a graph is known to be NP-hard. In random d-regular graphs, the problem of finding a maximum induced matching has been studied for d ∈ {3, 4, ..., 10 }. This was due to Duckworth et al.(2002) where they gave the asymptotically almost sure lower bounds and upper bonds on the size of maximum induced matchings in such graphs. The asymptotically almost sure lower bounds were achieved by analysing a degree-greedy algorithm using the differential equation method, whilst the asymptotically almost sure upper bounds were obtained by a direct expectation argument. In this paper, using the small subgraph conditioning method, we will show the asymptotically almost sure existence of an induced matching of certain size in random d-regular graphs, for d ∈ {3,4, 5}. This result improves the known asymptotically almost sure lower bound obtained by Duckworth et al.(2002).

Keywords

Regular Graph Interval Graph Chordal Graph Root Vertex Discrete Apply Mathematic 
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.

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References

  1. 1.
    Assiyatun, H.: Large Subgraphs of Regular Graphs, Doctoral Thesis, Department of Mathematics and Statistics, The University of Melbourne, Australia (2001)Google Scholar
  2. 2.
    Assiyatun, H., Duckworth, W.: Small Maximal Matchings of Random Cubic Graph (preprint)Google Scholar
  3. 3.
    Beis, M., Duckworth, W., Zito, M.: Packing Edges in Random Regular Graphs. In: Diks, K., Rytter, W. (eds.) MFCS 2002. LNCS, vol. 2420, pp. 118–130. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  4. 4.
    Bollobás, B.: Random Graphs. In: Temperley, H.N.V. (ed.) Combinatorics. London Mathematical Society Lecture Note Series, vol. 52, pp. 80–102. Cambridge University Press, Cambridge (1981)CrossRefGoogle Scholar
  5. 5.
    Cameron, K.: Induced Matchings. Discrete Applied Mathematics 24, 97–102 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Duckworth, W., Wormald, N.C., Zito, M.: Maximum Induced Matchings of Random Cubic Graph. The Journal of Computational and Applied Mathematics 142(1), 39–50 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Duckworth, W., Manlove, D., Zito, M.: On the Approximability of the Maximum Induced Matching Problem, Technical Report, TR-2000-56, Department of Computing Science of Glasgow University (2000)Google Scholar
  8. 8.
    Garmo, H.: Random Railways and Cycles in Random Regular Graphs, Doctoral Thesis, Uppsala University, Sweden (1998)Google Scholar
  9. 9.
    Golumbic, M.C., Lewenstein, M.: New results on induced matchings. Discrete Applied Mathematics 101, 157–165 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Golumbic, M.C., Laskar, R.C.: Irredundancy in Circular Arc Graphs. Discrete Applied Mathematics 44, 79–89 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Robalewska, H.D.: 2-Factors in Random Regular Graphs. Journal of Graph Theory 23(3), 215–224 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Janson, S.: Random Regular Graphs: Asymptotic Distributions and Contiguity. Combinatorics, Probability and Computing 4(4), 369–405 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Robinson, R.W., Wormald, N.C.: Almost All Regular Graphs are Hamiltonian. Random Structures and Algorithms 5(2), 363–374 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Robinson, R.W., Wormald, N.C.: Almost All Cubic Graphs are Hamiltonian. Random Structures & Algorithms 3, 117–125 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Stockmeyer, L.J., Vazirani, V.V.: NP-Completeness of Some Generalizations of the Maximum Matching Problem. Information Processing Letters 15(1), 14–19 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Wormald, N.C.: Models of Random Regular Graphs. In: Surveys in Combinatorics, pp. 239–298. Cambridge University Press, Cambridge (Canterbury 1999)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Hilda Assiyatun
    • 1
  1. 1.Department of MathematicsInstitut Teknologi BandungBandungIndonesia

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