On star-critical and upper size Ramsey numbers

Abstract

In this paper, we study the upper size Ramsey number $u(G_1,G_2)$, defined by Erdős and Faudree, as well as the star-critical Ramsey number $r_{\ast }(G_1,G_2)$, 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 $u(n{K}_k,m{K}_l)$ and $r_{\ast }(n{K}_k,m{K}_l)$ for $k,l\ge 3$ and large $m,n$; $u(C_n,C_m)$ for $m$ odd, with $n>m\ge 3$; and $r_{\ast }(C_n,C_m)$ for $m$ odd, with $n\ge m\ge 3$ and $(m,n)\ne (3,3)$.