On the sum of the reciprocals of cycle lengths in sparse graphs

Abstract

For a graphG let ℒ(G)=Σ{1/k contains a cycle of lengthk}. Erdős and Hajnal [1] introduced the real functionf(α)=inf {ℒ (G)|E(G)|/|V(G)|≧α} and suggested to study its properties. Obviouslyf(1)=0. We provef (k+1/k)≧(300k logk)−1 for all sufficiently largek, showing that sparse graphs of large girth must contain many cycles of different lengths.

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References

  1. [1]

    P. Erdős, Some recent progress on extremal problems in graph theory,Proc. 6th S. E. Conference on graph theory, Utilitas Math. 1975, 3–14.

  2. [2]

    A. Gyárfás, J. Komlós andE. Szemerédi, On the distribution of cycle lengths in graphs,J. Graph Theory,4 (1984), 441–462.

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Gyárfás, A., Prömel, H.J., Voigt, B. et al. On the sum of the reciprocals of cycle lengths in sparse graphs. Combinatorica 5, 41–52 (1985). https://doi.org/10.1007/BF02579441

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AMS subject classification (1980)

  • 05 C 38
  • 05 C 05