### Abstract

Given two graphs G_{1}, G_{2}, the Ramsey number R(G_{1}, G_{2}) is the smallest integer m such that, for any partition (E_{1}, E_{2}) of the edges of K_{m}, either G_{1} is a subgraph of the graph induced by E_{1}, or G_{2} is a subgraph of the graph induced by E_{2}. We show that R(C_{n}, C_{n})=2n-1 if n is odd, R(C_{n}, C_{2r-1})=2n-1 if n>r(2r-1), R(C_{n}, C_{2r})=n+r-1 if n>4r^{2}-r+2, R(C_{n}, K_{r})≤nr^{2} for all r,n, R(C_{n}, K_{r})=(r-1)(n-1)+1 if n≥r^{2}-2, R(C_{n}, K^{t+1}_{r})=t(n-1)+r for large n.

Original language | English |
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Pages (from-to) | 46-54 |

Number of pages | 9 |

Journal | Journal of Combinatorial Theory, Series B |

Volume | 14 |

Issue number | 1 |

DOIs | |

Publication status | Published - Feb 1973 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

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## Cite this

Bondy, J. A., & Erdös, P. (1973). Ramsey numbers for cycles in graphs.

*Journal of Combinatorial Theory, Series B*,*14*(1), 46-54. https://doi.org/10.1016/S0095-8956(73)80005-X