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

TL;DR

This paper determines the star critical Ramsey numbers for large cycles versus complete graphs, expanding understanding of edge colorings and Ramsey theory for specific graph pairs.

Contribution

It specifically computes $r_*(C_n, K_m)$ for $m \u2265 7$ and $n ; (m-3)(m-1)$, a novel extension in Ramsey number research.

Findings

Calculated $r_*(C_n, K_m)$ for specified parameters.

Extended known results to larger cycles and complete graphs.

Provided new bounds and exact values for star critical Ramsey numbers.

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 $K_n\rightarrow (G,H)$. The Ramsey number $r(G, H)$ is defined as the smallest positive integer $n$ such that $K_{n} \rightarrow (G, H)$. Star critical Ramsey $r_*(G, H)$ is defined as the largest value of $k$ such that $K_{r(G,H)-1} \sqcup K_{1,k} \rightarrow (G, H)$. In this paper, we find $r_*(C_n, K_m)$ for $m \geq 7$ and $n \geq (m-3)(m-1)$.