On Star critical Ramsey numbers related to large cycles versus complete graphs

Abstract

Let $K_n$ denote the complete graph on $n$ vertices and $G,H$ be finite graphs. Consider a two-coloring of edges of $K_n$. When a copy of $G$ in the first color, red, or a copy of $H$ in the second color, blue is in $K_n$, we write $K_n\to (G,H)$. The Ramsey number $r(G,H)$ is defined as the smallest positive integer $n$ such that $K_n\to (G,H)$. Star critical Ramsey $r_{\ast }(G,H)$ is defined as the largest value of $k$ such that $K_{r(G,H)-1}\bigsqcup K_{1,k}\to (G,H)$. In this paper, we find $r_{\ast }(C_n,K_m)$ for $m\ge 7$ and $n\ge (m-3)(m-1)$.

10 pages, 2 figures. arXiv admin note: text overlap with arXiv:1902.02646