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Cycles are strongly Ramsey-unsaturated

Skokan, Jozef and Stein, M. (2014) Cycles are strongly Ramsey-unsaturated. Combinatorics, Probability and Computing, 23 (04). pp. 607-630. ISSN 0963-5483

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Identification Number: 10.1017/S0963548314000212


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
Official URL:
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
Sets: Departments > Mathematics
Date Deposited: 24 Jun 2014 08:58
Last Modified: 20 Feb 2021 05:16
Projects: 11090141
Funders: Fondecyt

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