### Abstract

Given two graphs G1 and G2, the Ramsey number R(G1,G2) is the
smallest integer N such that, for any graph G of order N, either G1 is a subgraph of G, or G2 is a subgraph of the complement of G. Let Cm denote a cycle of order m, K1,n a star of order n + 1 and Wn a wheel of order n + 1. Already back in the 1970s, exact values of the Ramsey numbers R(Cm,K1,n) have been determined for all m ≥ 2n and for all odd m ≤ 2n − 1, but for even m < 2n not many exact values are known. In this paper, we use a result of Bondy on pancyclicity to fill a considerable part of this gap. We show that R(Cm,K1,n) = 2n for even m with n < m < 2n, and that R(Cm,K1,n) = 2m−1 for even m with 3n/4+1 ≤ m ≤ n. In addition, we determine another six formerly unknown exact values of Ramsey numbers, namely R(C6,K1,n) for 7 ≤ n ≤ 11, and R(C6,W9).

Original language | Undefined |
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Pages (from-to) | 481-493 |

Number of pages | 13 |

Journal | Journal of combinatorics (Somerville) |

Volume | 7 |

Issue number | 2-3 |

DOIs | |

Publication status | Published - Feb 2016 |

### Keywords

- EWI-26877
- MSC-05C
- Cycle
- IR-100080
- Star
- METIS-316849
- Ramsey number

## Cite this

Zhang, Y., Broersma, H. J., & Chen, Y. (2016). Narrowing down the gap on cycle-star Ramsey numbers.

*Journal of combinatorics (Somerville)*,*7*(2-3), 481-493. https://doi.org/10.4310/JOC.2016.v7.n2.a13