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Discussiones Mathematicae Graph Theory 25(1-2) (2005) 129-139
DOI: https://doi.org/10.7151/dmgt.1267

THE CYCLE-COMPLETE GRAPH RAMSEY NUMBER R(C₅,K₇)

Ingo Schiermeyer

Institut für Diskrete Mathematik und Algebra
Technische Universität Bergakademie Freiberg
09596 Freiberg, Germany
e-mail: schierme@tu-freiberg.de

Abstract

The cycle-complete graph Ramsey number r(C_m,K_n) is the smallest integer N such that every graph G of order N contains a cycle C_m on m vertices or has independence number α(G) ≥ n. It has been conjectured by Erdős, Faudree, Rousseau and Schelp that r(C_m,K_n) = (m−1)(n−1)+1 for all m ≥ n ≥ 3 (except r(C₃,K₃) = 6). This conjecture holds for 3 ≤ n ≤ 6. In this paper we will present a proof for r(C₅,K₇) = 25.

Keywords: Ramsey numbers, extremal graphs.

2000 Mathematics Subject Classification: 05C55, 05C35.

References

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Received 6 November 2003
Revised 16 February 2005