Ramsey numbers of color critical graphs versus large generalized fans

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 $K_{N-1}\bigsqcup K_{1,k}$ 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} $r_{\ast }(G,H)$ as the smallest integer $k$ such that every 2-coloring of the edges of $K_{N-1}\bigsqcup K_{1,k}$ contains either a red $G$ or a blue $H$, where $N=R(G,H)$. For sufficiently large $n$, Li and Rousseau~(1996) proved that $R(K_{k+1},K_1+n{K}_t)=knt+1$, Hao, Lin~(2018) showed that $r_{\ast }(K_{k+1},K_1+n{K}_t)=(k-1)tn+t$; Li and Liu~(2016) proved that $R(C_{2k+1},K_1+n{K}_t)=2nt+1$, and Li, Li, and Wang~(2020) showed that $r_{\ast }(C_{2m+1},K_1+n{K}_t)=nt+t$. 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 $k\ge 2$, $t\ge 2$ and $n$ is sufficiently large, $R(G,K_1+n{K}_t)=knt+1$ and $r_{\ast }(G,K_1+n{K}_t)=(k-1)nt+t$.