  For graphs $G$, $F$ and $H$, let $G\rightarrow (F,H)$ signify that any edge
coloring of $G$ in red and blue contains a red $F$ or a blue $H$. The Ramsey
number $R(F,H)=\min\{r|\; K_r\rightarrow (F,H)\}$. In this note, we consider
redundant edges in Ramsey graphs, which are associate with critical Ramsey
numbers. For an integer $k\ge 1$, let ${\mathbb G}=\{G_k,G_{k+1},\dots \}$ be a
class of graphs with $\delta(G_n)\ge 1$. We define the critical Ramsey number
$R_{\mathbb G}(F,H)$ with respect to $\mathbb G$ to be $\max\{n|\; K_r\setminus
G_n \rightarrow(F,H),\,G_n\in{\mathbb G}\big\}$, where $r=R(F,H)$. We shall
determine some $R_{\mathbb G}(F,H)$, where ${\mathbb G}$ consists of stars,
matchings and complete graphs, respectively.
