×

The Ramsey number for a cycle of length six versus a clique of order eight. (English) Zbl 1159.05033

It was conjectured by P. Erdős, R.J. Faudree, C.C. Rousseau, and R.H. Schelp in [J. Graph Theory 2 , 53-64 (1978; Zbl 0383.05027)] that \(r(K_m, C_n) = (m - 1)(n - 1) + 1\) for \(3 \leq m \leq n\) and \((m,n) \neq (3,3)\), and the question was posed to find the smallest \(n\) for which this is true. It is shown that the Ramsey number \(r(K_8, C_6) = 36 = (8 - 1)(6 - 1) + 1\). The conjecture has been verified for \(m \leq 7\), and this adds to the results that support the conjecture of Erdős et.al.

MSC:

05C55 Generalized Ramsey theory
05C38 Paths and cycles
05C35 Extremal problems in graph theory

Citations:

Zbl 0383.05027
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bollobás, B.; Jayawardene, C. J.; Yang, J. S.; Huang, Y. R.; Rousseau, C. C.; Zhang, K. M., On a conjecture involving cycle-complete graph Ramsey numbers, Australasian Journal of Combinatorics, 22, 63-71 (2000) · Zbl 0963.05094
[2] Bondy, J. A.; Murty, U. S.R., Graph Theory with Applications (1976), Macmillan: Macmillan London, (Elsevier, New York) · Zbl 1134.05001
[3] Chen, Y. J.; Cheng, T. C.E.; Zhang, Y. Q., European Journal of Combinatorics, 29, 1337-1352 (2008)
[4] Cheng, T. C.E.; Chen, Y. J.; Zhang, Y. Q.; Ng, C. T., The Ramsey numbers for a cycle of length six or seven versus a clique of order seven, Discrete Mathematics, 307, 1047-1053 (2007) · Zbl 1120.05059
[5] Erdös, P.; Faudree, R. J.; Rousseau, C. C.; Schelp, R. H., On cycle-complete graph Ramsey numbers, Journal of Graph Theory, 2, 53-64 (1978) · Zbl 0383.05027
[6] Faudree, R. J.; Schelp, R. H., All Ramsey numbers for cycles in graphs, Discrete Mathematics, 8, 313-329 (1974) · Zbl 0294.05122
[7] McKay, B. D.; Zhang, K. M., The value of the Ramsey number \(R(3, 8)\), Journal of Graph Theory, 16, 99-105 (1992) · Zbl 0758.05098
[8] Radziszowski, S. P., Small Ramsey numbers, Electronic Journal of Combinatorics, 1 (2006), DS1.11
[9] Rosta, V., On a Ramsey type problem of J.A. Bondy and P. Erdös, I & II, Journal of Combinatorial Theory, Ser. B, 15, 94-120 (1973) · Zbl 0261.05119
[10] Schiermeyer, I., All cycle-complete graph Ramsey numbers \(r(C_m, K_6)\), Journal of Graph Theory, 44, 251-260 (2003) · Zbl 1031.05086
[11] Yang, J. S.; Huang, Y. R.; Zhang, K. M., The value of the Ramsey number \(R(C_n, K_4)\) is \(3(n - 1) + 1\), Australasian Journal of Combinatorics, 20, 205-206 (1999) · Zbl 0931.05057
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.