Ramsey Numbers for Complete Graphs Versus Generalized Fans

Abstract

For two graphs G and H, let r(G, H) and \(r_*(G,H)\) denote the Ramsey number and star-critical Ramsey number of G versus H, respectively. In 1996, Li and Rousseau proved that \(r(K_{m},F_{t,n})=tn(m-1)+1\) for \(m\ge 3\) and sufficiently large n, where \(F_{t,n}=K_{1}+nK_{t}\). Recently, Hao and Lin proved that \(r(K_{3},F_{3,n})=6n+1\) for \(n\ge 3\) and \(r_{*}(K_{3},F_{3,n})=3n+3\) for \(n\ge 4\). In this paper, we show that \(r(K_{m}, sF_{t,n})=tn(m+s-2)+s\) for sufficiently large n and, in particular, \(r(K_{3}, sF_{t,n})=tn(s+1)+s\) for \(t\in \{3,4\},n\ge t\) and \(s\ge 1\). We also show that \(r_{*}(K_{3}, F_{4,n})=4n+4\) for \(n\ge 4\) and establish an upper bound for \(r(F_{2,m},F_{t,n})\).