On star-critical and upper size Ramsey numbers

Abstract

In this paper, we study the upper size Ramsey number u(G1,G2), defined by Erdős and Faudree, as well as the star-critical Ramsey number r∗(G1,G2), defined by Hook and Isaak. We define Ramsey-full graphs and size Ramsey good graphs, and perform a detailed study on these graphs. We generalize earlier results by determining u(nKk,mKl) and r∗(nKk,mKl) for k,l≥3 and large m,n; u(Cn,Cm) for m odd, with n>m≥3; and r∗(Cn,Cm) for m odd, with n≥m≥3 and (m,n)≠(3,3).