The star-critical Ramsey number r∗(H1, H2) is the smallest integer k such that every red/blue coloring of the edges of Kn − K1,n−k−1 contains either a red copy of H1 or a blue copy of H2, where n is the graph Ramsey number R(H1, H2). We study the cases of r∗(C4, Cn) and R(C4, Wn). In particular, we prove that r∗(C4, Cn) = 5 for all n \u3e 4, obtain a general characterization of Ramsey-critical (C4, Wn)-graphs, and establish the exact values of R(C4, Wn) for 9 cases of n between 18 and 44
