Abstract
Let \(ex(n, C_4)\) denote the maximum size of a \(C_4\)-free graph of order \(n\). For an even integer or odd prime power \(q\), we prove that \(ex(q^2+q+2,C_4)<\frac{1}{2}(q+1)(q^2+q+2)\), which leads to an improvement of the upper bound on Ramsey numbers \(R(C_4,W_{q^2+2})\), where \(W_n\) is a wheel of order \(n\). By using a simple polarity graph \(G_q\) for a prime power \(q\), we construct the graphs whose complements do not contain \(K_{1,m}\) or \(W_m\), and then determine some exact values of \(R(C_4,K_{1,m})\) and \(R(C_4,W_{m})\). In particular, we prove that \(R(C_4,K_{1, q^2-2})=q^2+q-1\) for \(q\ge 3\), \(R(C_4,W_{q^2-1})=q^2+q-1\) for \(q\ge 5\), and \(R(C_4,W_{q^2+2})=q^2+q+2\) for \(q\ge 7\).
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Supported by NSFC (60973011, 61272004), and Fundamental Research Funds for Central Universities. Supported by Polish National Science Centre Grant 2011/02/A/ST6/00201.
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Wu, Y., Sun, Y., Zhang, R. et al. Ramsey Numbers of \(C_4\) versus Wheels and Stars. Graphs and Combinatorics 31, 2437–2446 (2015). https://doi.org/10.1007/s00373-014-1504-3
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DOI: https://doi.org/10.1007/s00373-014-1504-3