On Ramsey Minimal Graphs

Abstract

A graph G is r-Ramsey-minimal with respect to a graph H if every r-coloring of the edges of G yields a monochromatic copy of H, but the same is not true for any proper subgraph of G. In this paper we show that for any integer $k \geq 3$ and $r \geq 2$, there exists a constant $c>1$ such that for large enough n, there exist at least $c^{n^2}$ nonisomorphic graphs on at most n vertices, each of which is r-Ramsey-minimal with respect to the complete graph $K_k$. Furthermore, in the case $r=2$, we give an asymmetric version of the above result.