Abstract
For given graphs G 1 and G 2, the Ramsey number R(G 1, G 2) is the least integer n such that every 2-coloring of the edges of K n contains a subgraph isomorphic to G 1 in the first color or a subgraph isomorphic to G 2 in the second color. Surahmat et al. proved that the Ramsey number \({R(C_4, W_n) \leq n + \lceil (n-1)/3\rceil}\). By using asymptotic methods one can obtain the following property: \({R(C_4, W_n) \leq n + \sqrt{n}+o(1)}\). In this paper we show that in fact \({R(C_4, W_n) \leq n + \sqrt{n-2}+1}\) for n ≥ 11. Moreover, by modification of the Erdős-Rényi graph we obtain an exact value \({R(C_4, W_{q^2+1}) = q^2 + q + 1}\) with q ≥ 4 being a prime power. In addition, we provide exact values for Ramsey numbers R(C 4, W n ) for 14 ≤ n ≤ 17.
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This research was funded by the Polish National Science Centre (contract number DEC-2012/05/N/ST6/03063).
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Dybizbański, J., Dzido, T. On Some Ramsey Numbers for Quadrilaterals Versus Wheels. Graphs and Combinatorics 30, 573–579 (2014). https://doi.org/10.1007/s00373-013-1293-0
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DOI: https://doi.org/10.1007/s00373-013-1293-0