Given two graphs $G$ and $H$, the { Ramsey number} $R(G,H)$ is the smallest positive integer $N$ such that every 2-coloring of the edges of $K_{ N}$ contains either a red $G$ or a blue $H$. Let $K_{ N-1}\sqcup K_{ 1,k}$ be the graph obtained from $K_{ N-1}$ by adding a new vertex $v$ connecting $k$ vertices of $K_{ N-1}$. Hook and Isaak (2011) defined the { \em star-critical Ramsey number} $r_{ *}(G,H)$ as the smallest integer $k$ such that every 2-coloring of the edges of $K_{ N-1}\sqcup K_{ 1,k}$ contains either a red $G$ or a blue $H$, where $N=R(G, H)$. For sufficiently large $n$, Li and Rousseau~(1996) proved that $R(K_{ k+1},K_{ 1}+nK_{ t})=knt +1$, Hao, Lin~(2018) showed that $r_{ *}(K_{ k+1},K_{ 1}+nK_{ t})=(k-1)tn+t$; Li and Liu~(2016) proved that $R(C_{ 2k+1}, K_{ 1}+nK_{ t})=2nt+1$, and Li, Li, and Wang~(2020) showed that $r_{ *}(C_{ 2m+1},K_{ 1}+nK_{ t})=nt+t$. A graph $G$ with $\chi(G)=k+1$ is called edge-critical if $G$ contains an edge $e$ such that $\chi(G-e)=k$. In this paper, we extend the above results by showing that for an edge-critical graph $G$ with $\chi(G)=k+1$, when $k\geq 2$, $t\geq 2$ and $n$ is sufficiently large, $R(G, K_{ 1}+nK_{ t})=knt+1$ and $r_{ *}(G,K_{ 1}+nK_{ t})=(k-1)nt+t$.
Categories: math.CO