The cycle-complete graph Ramsey number r(Cₘ,Kₙ) is the smallest integer N such that every graph G of order N contains a cycle Cₘ on m vertices or has independence number α(G) ≥ n. It has been conjectured by Erdős, Faudree, Rousseau and Schelp that r(Cₘ,Kₙ) = (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.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1267,
author = {Ingo Schiermeyer},
title = {The cycle-complete graph Ramsey number r(C5,K7)},
journal = {Discussiones Mathematicae Graph Theory},
volume = {25},
year = {2005},
pages = {129-139},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1267}
}
Ingo Schiermeyer. The cycle-complete graph Ramsey number r(C₅,K₇). Discussiones Mathematicae Graph Theory, Tome 25 (2005) pp. 129-139. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1267/
[000] [1] J.A. Bondy and P. Erdős, Ramsey numbers for cycles in graphs, J. Combin. Theory (B) 14 (1973) 46-54, doi: 10.1016/S0095-8956(73)80005-X. | Zbl 0248.05127
[001] [2] B. Bollobás, C.J. Jayawardene, Z.K. Min, C.C. Rousseau, H.Y. Ru and J. Yang, On a conjecture involving cycle-complete graph Ramsey numbers, Australas. J. Combin. 22 (2000) 63-72. | Zbl 0963.05094
[002] [3] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (Macmillan, London and Elsevier, New York, 1976). | Zbl 1226.05083
[003] [4] V. Chvátal and P. Erdős, A note on hamiltonian circuits, Discrete Math. 2 (1972) 111-113, doi: 10.1016/0012-365X(72)90079-9. | Zbl 0233.05123
[004] [5] P. Erdős, R.J. Faudree, C.C. Rousseau and R.H. Schelp, On cycle-complete graph Ramsey numbers, J. Graph Theory 2 (1978) 53-64, doi: 10.1002/jgt.3190020107. | Zbl 0383.05027
[005] [6] R.J. Faudree and R.H. Schelp, All Ramsey numbers for cycles in graphs, Discrete Math. 8 (1974) 313-329, doi: 10.1016/0012-365X(74)90151-4. | Zbl 0294.05122
[006] [7] C.J. Jayawardene and C.C. Rousseau, The Ramsey number for a qudrilateral versus a complete graph on six vertices, Congr. Numer. 123 (1997) 97-108. | Zbl 0902.05050
[007] [8] C.J. Jayawardene and C.C. Rousseau, The Ramsey Number for a Cycle of Length Five vs. a Complete Graph of Order Six, J. Graph Theory 35 (2000) 99-108, doi: 10.1002/1097-0118(200010)35:2<99::AID-JGT4>3.0.CO;2-6 | Zbl 0969.05044
[008] [9] V. Nikiforov, The cycle-complete graph Ramsey numbers, preprint 2003, Univ. of Memphis. | Zbl 1071.05051
[009] [10] S.P. Radziszowski, Small Ramsey numbers, Elec. J. Combin. 1 (1994) DS1.
[010] [11] S.P. Radziszowski and K.-K. Tse, A Computational Approach for the Ramsey Numbers R(C₄,Kₙ), J. Comb. Math. Comb. Comput. 42 (2002) 195-207.
[011] [12] V. Rosta, On a Ramsey Type Problem of J.A. Bondy and P. Erdős, I & II, J. Combin. Theory (B) 15 (1973) 94-120, doi: 10.1016/0095-8956(73)90035-X.
[012] [13] I. Schiermeyer, All Cycle-Complete Graph Ramsey Numbers r(Cₘ, K₆), J. Graph Theory 44 (2003) 251-260, doi: 10.1002/jgt.10145.
[013] [14] Y.J. Sheng, H.Y. Ru and Z.K. Min, The value of the Ramsey number R(Cₙ,K₄) is 3(n-1) +1 (n ≥ 4), Australas. J. Combin. 20 (1999) 205-206. | Zbl 0931.05057
[014] [15] A. Thomason, private communication.