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The Ramsey number \(R(C_8,K_8)\). (English) Zbl 1169.05026

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 the Ramsey number \(R(C_m, K_n) = (m - 1)(n - 1) + 1\) for \(m \geq n\) except for \((m, n) = (3, 3)\). The Conjecture has been verified for \(m \leq 7\). It is shown that \(R(C_8, K_8) = 50\), which is a start on the verification of the Conjecture for \(m = 8\).

MSC:

05C55 Generalized Ramsey theory

Citations:

Zbl 0383.05027
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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, London and Elsevier: Macmillan, London and Elsevier New York · Zbl 1134.05001
[3] Y.J. Chen, T.C.E. Cheng, Y.Q. Zhang, The Ramsey numbers \(R ( C_m , K_7 )R ( C_7 , K_8 )\); Y.J. Chen, T.C.E. Cheng, Y.Q. Zhang, The Ramsey numbers \(R ( C_m , K_7 )R ( C_7 , K_8 )\)
[4] Edwin Cheng, T. C.; 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] Radziszowski, S. P., Small Ramsey numbers, Electronic Journal of Combinatorics, 1, DS1.10 (2004)
[8] Rosta, V., On a Ramsey type problem of J. A. Bondy and P. Erdös, I & II, Journal of Combinatorial Theory, Series B, 15, 94-120 (1973) · Zbl 0261.05119
[9] Schiermeyer, I., All cycle-complete graph Ramsey numbers \(r(C_m, K_6)\), Journal of Graph Theory, 44, 251-260 (2003) · Zbl 1031.05086
[10] 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
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