Skokan, Jozef ORCID: 0000-0003-3996-7676 and Stein, M.
(2014)
*Cycles are strongly Ramsey-unsaturated.*
Combinatorics, Probability and Computing, 23 (04).
pp. 607-630.
ISSN 0963-5483

## Abstract

We call a graph H Ramsey-unsaturated if there is an edge in the complement of H such that the Ramsey number r(H) of H does not change upon adding it to H. This notion was introduced by Balister, Lehel and Schelp in [J. Graph Theory 51 (2006), pp. 22–32], where it is shown that cycles (except for C 4) are Ramsey-unsaturated, and conjectured that, moreover, one may add any chord without changing the Ramsey number of the cycle Cn , unless n is even and adding the chord creates an odd cycle. We prove this conjecture for large cycles by showing a stronger statement. If a graph H is obtained by adding a linear number of chords to a cycle Cn , then r(H)=r(Cn), as long as the maximum degree of H is bounded, H is either bipartite (for even n) or almost bipartite (for odd n), and n is large. This motivates us to call cycles strongly Ramsey-unsaturated. Our proof uses the regularity method.

Item Type: | Article |
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Official URL: | http://journals.cambridge.org/action/displayAbstra... |

Additional Information: | © 2014 Cambridge University Press 2014 |

Divisions: | Mathematics |

Subjects: | Q Science > QA Mathematics Q Science > QA Mathematics > QA75 Electronic computers. Computer science |

JEL classification: | C - Mathematical and Quantitative Methods > C0 - General > C02 - Mathematical Methods C - Mathematical and Quantitative Methods > C6 - Mathematical Methods and Programming > C61 - Optimization Techniques; Programming Models; Dynamic Analysis |

Date Deposited: | 24 Jun 2014 08:58 |

Last Modified: | 14 Sep 2024 06:33 |

Projects: | 11090141 |

Funders: | Fondecyt |

URI: | http://eprints.lse.ac.uk/id/eprint/57212 |

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