  For graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the smallest $r$ such
that any red-blue edge coloring of $K_r$ contains a red $G$ or a blue $H$. The
path-critical Ramsey number $R_{\pi}(G,H)$ is the largest $n$ such that any
red-blue edge coloring of $K_r \setminus P_{n}$ contains a red $G$ or a blue
$H$, where $r=R(G,H)$ and $P_{n}$ is a path of order $n$. In this note, we show
a general upper bound for $R_{\pi}(G,H)$, and determine the exact values for
some cases of $R_{\pi}(G,H)$.
