Ramsey numbers of stars versus wheels of similar sizes (Q1773362)

The Ramsey number \(R(G,H)\) of the graphs \(G\) and \(H\) is the smallest positive integer \(r\) such that any red-blue colouring of the edges of \(K_r\) yields a red \(G\) or a blue \(H\). This paper considers the case of the wheel \(W_m\) on \(m+1\) vertices, and the star \(S_n\) on \(n\) vertices. Results of \textit{Surahmat, E. T. Baskoro} and \textit{H. J. Broersma} [University of Twende Memorandum No. 1621, March 2002] and \textit{Surahmat} and \textit{E. T. Baskoro} [Proc. 12th Australas. Workshop Comb. Algor., Bandung, Indonesia, July 14--17, 174--179 (2001)] show that \(R(W_4,S_n)=2n-1\) when \(n\geq 3\) and odd; and \(2n+1\) when \(n\geq4\) and even; and that \(R(W_m,S_n) =3n-2\) when \(m=5\) and \(n\geq 3\) and when \(m\geq 5\) and odd, and \(n\geq 2m-4\). In this paper this result is extended to the case where \(m\geq 7\) and odd and \(n=m\), \(m+1\), \(m+2\). Note that this completes the result for \(n\geq m=7\). A lower bound of \(2n+1\) is obtained in the case where \(n\geq m\geq 6\) and \(m\) is even.