  For two graphs $G$ and $H$, the Ramsey number $r(G,H)$ is the smallest
positive integer $r$, such that any red/blue coloring of the edges of the graph
$K_r$ contains either a red subgraph that is isomorphic to $G$ or a blue
subgraph that is isomorphic to $H$. Let $S_k=K_{1,k}$ be a star of order $k+1$
and $K_n\sqcup S_k$ be a graph obtained from $K_n$ by adding a new vertex $v$
and joining $v$ to $k$ vertices of $K_n$. The star-critical Ramsey number
$r_*(G,H)$ is the smallest positive integer $k$ such that any red/blue coloring
of the edges of graph $K_{r-1}\sqcup S_k$ contains either a red subgraph that
is isomorphic to $G$ or a blue subgraph that is isomorphic to $H$, where
$r=r(G,H)$. In this paper, it is shown that $r_*(F_n,K_4)=4n+2$, where
$n\geq{4}$.
