All Ramsey $(C_n,K_6)$ critical graphs for large $n$

Abstract

Let $G$ and $H$ be finite graphs. If for any two-coloring of the edges of a complete graph $K_n$, there is a copy of $G$ in the first color, red, or a copy of $H$ in the second color, blue, we will say $K_n\rightarrow (G,H)$. The Ramsey number $r(G, H)$ is defined as the smallest positive integer $n$ such that $K_{n} \rightarrow (G, H)$. A two-coloring of $K_{r(G, H)-1}$ such that $K_{r(G, H)-1} \not \rightarrow (G,H)$ is called a critical coloring. A Ramsey critical $r(G, H)$ graph is a graph induced by the first color of a critical coloring. In this paper, when $n \geq 15$, we show that there exist exactly sixty eight non-isomorphic Ramsey critical $r(C_n, K_6)$ graphs.