Abstract
For given graphs G and H, the Ramsey number R(G, H) is the smallest positive integer N such that for every graph F of order N the following holds: either F contains G as a subgraph or the complement of F contains H as a subgraph. In this paper, we determine the Ramsey number R(C n , W m ) = 3n − 2 for odd m ≥ 5 and \(n > \frac{5m-9}{2}\).
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Bondy, J.A.: Pancyclic graphs. J. Combin. Theory Ser. B 11, 80–84 (1971)
Brandt, S., Faudree, R.J., Goddard, W.: Weakly pancyclic graphs. J. Graph Theory 27, 141–176 (1998)
Burr, S.A., Erdős, P.: Generalization of a Ramsey-theoretic result of Chvátal. J. Graph Theory 7, 39–51 (1983)
Chvátal, V., Erdős, P.: A note on Hamiltonian circuits. Discrete Math. 2, 111–113 (1972)
Chvátal, V., Harary, F.: Generalized Ramsey theory for graphs, III. Small off-diagonal numbers. Pacific J. Math. 41, 335–345 (1972)
Dirac, G.: Some theorems on abstract graphs. Proc. London Math. Soc. 2, 69–81 (1952)
Faudree, R.J., Schelp, R.H.: All Ramsey numbers for cycles in graphs. Discrete Math. 8, 313–329 (1974)
Hendry, G.R.T.: Ramsey numbers for graphs with five vertices. J. Graph Theory 13, 181–203 (1989)
Jayawardene, C.J., Rousseau, C.C.: Ramsey number R(C5, G) for all graphs G of order six. Ars Combin. 57, 163–173 (2000)
Radziszowski, S.P.: Small Ramsey numbers. Electronic J. Combinatorics DS1 (2007)
Radziszowski, S.P., Xia, J.: Paths, cycles and wheels without antitriangles. Australasian J. Combin. 9, 221–232 (1994)
Rosta, V.: On a Ramsey type problem of J.A. Bondy and P. Erdős, I & II. J. Combinatorial Theory (B) 15, 94–120 (1973)
Surahmat, Baskoro, E.T., Broersma, H.J.: The Ramsey numbers of large cycles versus small wheels. Integer: The Electronic J. Combin. Number Theory 4, ♯ A10 (2004)
Surahmat, Baskoro, E.T., Nababan, S.M.: The Ramsey numbers for a cycle of length four versus a small wheel. Proceedings of the 11-th Conference Indonesian Mathematics, Malang, Indonesia, July 22–25 (2002) 172–178
Surahmat, Baskoro, E.T.: Ioan Tomescu, The Ramsey numbers of large cycles versus even wheels. Discrete Math. 306, 3334–3337 (2006)
Tse, K.K.: On the Ramsey number of the quadrilateral versus the book and the wheel. Australasian J. Combin. 27, 163–167 (2003)
Zhou, H.L.: The Ramsey number of an odd cycle with respect to a wheel (in Chinese). J. Math., Shuxu Zazhi (Wuhan) 15, 119–120 (1995)
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Surahmat, Ioan Tomescu: Part of the work was done while the first and the last authors were visiting the School of Mathematical Sciences, Government College University, Lahore, Pakistan.
Surahmat: Research partially support under TWAS, Trieste, Italy, RGA No: 06-018 RG/MATHS/AS–UNESCO FR: 3240144875.
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Surahmat, Baskoro, E.T. & Tomescu, I. The Ramsey Numbers of Large cycles Versus Odd Wheels. Graphs and Combinatorics 24, 53–58 (2008). https://doi.org/10.1007/s00373-007-0764-6
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DOI: https://doi.org/10.1007/s00373-007-0764-6