The Ramsey numbers \(R(C_m,K_7)\) and \(R(C_7,K_8)\). (English) Zbl 1154.05044

The Ramsey number \(R(G_1,G_2)\) is the smallest integer \(n\) such that for any \(G\) of order \(n\) either \(G\) contains \(G_1\) or the complement of \(G\) contains \(G_2\). In the paper it is shown that \(R(C_m,K_7) = 6m-5\) for \(m \geq 7\) and \(R(C_7,K_8) = 43\). These results confirm the conjecture of P. Erdős, R.J. Faudree, C.C. Rousseau, and R.H. Schelp [J. Graph Theory 2, 53–64 (1978; Zbl 0383.05027)].


05C55 Generalized Ramsey theory


Zbl 0383.05027
Full Text: DOI


[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] 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
[4] Chvátal, V.; Erdös, P., A note on hamiltonian circuits, Discrete Mathematics, 2, 111-113 (1972) · Zbl 0233.05123
[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] Ore, O., Note on Hamilton circuits, American Mathematical Monthly, 67, 55 (1960) · Zbl 0089.39505
[8] Radziszowski, S. P., Small Ramsey numbers, Electronic Journal of Combinatorics, DS1.11 (2006)
[9] 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
[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.