Wheel and star-critical Ramsey numbers for quadrilateral

The star-critical Ramsey number r∗(H1,H2)r∗(H1,H2) is the smallest integer kk such that every red/blue coloring of the edges of Kn−K1,n−k−1Kn−K1,n−k−1 contains either a red copy of H1H1 or a blue copy of H2H2, where nn is the graph Ramsey number R(H1,H2)R(H1,H2). We study the cases of r∗(C4,Cn)r∗(C4,Cn) and R(C4,Wn)R(C4,Wn). In particular, we prove that r∗(C4,Cn)=5r∗(C4,Cn)=5 for all n≥4n≥4, obtain a general characterization of Ramsey-critical (C4,Cn)(C4,Cn)-graphs, and establish the exact values of R(C4,Wn)R(C4,Wn) for 9 cases of nn between 1818 and 4444.