Some Ramsey Numbers of Graphs with Bridge

It is shown that if \(H\) is a connected graph obtained from \(H_1\) and \(H_2\) by joining them with a bridge, then \(r(K_k, H) \leq r(K_k, H_1) + r(K_k, H_2) + k – 2\). We give some applications of this result.