### Abstract

For a graph G let ℒ(G)=Σ{1/k contains a cycle of length k}. Erdo{combining double acute accent}s and Hajnal [1] introduced the real function f(α)=inf {ℒ (G)|E(G)|/|V(G)|≧α} and suggested to study its properties. Obviously f(1)=0. We prove f (k+1/k)≧(300 k log k)^{-1} for all sufficiently large k, showing that sparse graphs of large girth must contain many cycles of different lengths.

Original language | English |
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Pages (from-to) | 41-52 |

Number of pages | 12 |

Journal | Combinatorica |

Volume | 5 |

Issue number | 1 |

DOIs | |

Publication status | Published - Mar 1 1985 |

### Keywords

- AMS subject classification (1980): 05C38, 05C05

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Computational Mathematics

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## Cite this

Gyárfás, A., Prömel, H. J., Voigt, B., & Szemerédi, E. (1985). On the sum of the reciprocals of cycle lengths in sparse graphs.

*Combinatorica*,*5*(1), 41-52. https://doi.org/10.1007/BF02579441