Star-critical Ramsey number of $K_4$ versus $F_n$

TL;DR

This paper determines the exact star-critical Ramsey number for the pair of graphs involving a star and a complete graph, specifically proving that it equals 4n+2 for all n ≥ 4.

Contribution

It establishes the precise value of the star-critical Ramsey number for $F_n$ versus $K_4$, extending understanding of Ramsey theory for these graph classes.

Findings

Proves $r_*(F_n,K_4)=4n+2$ for all $n geq 4$

Provides a new exact value for a specific star-critical Ramsey number

Enhances the theoretical framework of Ramsey numbers involving stars and complete graphs

Abstract

For two graphs $G$ and $H$, the Ramsey number $r(G,H)$ is the smallest positive integer $r$, such that any red/blue coloring of the edges of the graph $K_r$ contains either a red subgraph that is isomorphic to $G$ or a blue subgraph that is isomorphic to $H$. Let $S_k=K_{1,k}$ be a star of order $k+1$ and $K_n\sqcup S_k$ be a graph obtained from $K_n$ by adding a new vertex $v$ and joining $v$ to $k$ vertices of $K_n$. The star-critical Ramsey number $r_*(G,H)$ is the smallest positive integer $k$ such that any red/blue coloring of the edges of graph $K_{r-1}\sqcup S_k$ contains either a red subgraph that is isomorphic to $G$ or a blue subgraph that is isomorphic to $H$, where $r=r(G,H)$. In this paper, it is shown that $r_*(F_n,K_4)=4n+2$, where $n\geq{4}$.