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Large holes in sparse random graphs

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Abstract

We show that there is a function α(r) such that for each constantr≧3, almost everyr-regular graph onn vertices has a hole (vertex induced cycle) of size at least α(r)n asn→∞. We also show that there is a function β(c) such that forc>0 large enough,G n, p ,p=c/n almost surely has a hole of size at least β(c)n asn→∞.

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References

  1. B. Bollobás, A probablistic proof of an asymptotic formula for the number of labelled regular graphs,European Journal of Combinatorics 1 (1980), 311–316.

    MATH  MathSciNet  Google Scholar 

  2. B. Bollobás, T. I. Fenner andA. M. Frieze, Long cycles in sparse random graphs,Graph Theory and Combinatorics, A volume in honour of Paul Erdős, (ed. B. Bollobás), Academic Press, London (1984), 59–64.

    Google Scholar 

  3. P. Erdős andZ. Palka, Trees in random graphs,Discrete Mathematics 46 (1983), 145–150.

    Article  MathSciNet  Google Scholar 

  4. W. Fernandez de la Vega, Induced trees in sparse random graphs,to appear.

  5. A. M. Frieze andB. Jackson, Large induced trees in sparse random graphs,Journal of Combinatorial B42 (1987), 181–195.

    MathSciNet  Google Scholar 

  6. W. Hoeffding, Probability inequalities for sums of bounded random variables,Journal of the American Statistical Association 58 (1963), 13–30.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Queen Mary College, London University, England

    A. M. Frieze

  2. Carnegie Mellon University, Pittsburg, USA

    A. M. Frieze

  3. Goldsmiths College, London University, England

    B. Jackson

Authors
  1. A. M. Frieze
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  2. B. Jackson
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Frieze, A.M., Jackson, B. Large holes in sparse random graphs. Combinatorica 7, 265–274 (1987). https://doi.org/10.1007/BF02579303

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  • DOI: https://doi.org/10.1007/BF02579303

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