On some path-critical Ramsey numbers

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 \(R_{\pi }(G,H)\).