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 $r(F_{2,m},F_{t,n})$.