A note on the Ramsey number of even wheels versus stars

TL;DR

This paper determines the Ramsey number for even wheels versus stars, establishing tight bounds and exact values for large even n, advancing understanding of graph Ramsey theory.

Contribution

It provides new bounds and the exact value of the Ramsey number R(W_n, S_n) for even n, refining previous results in graph Ramsey theory.

Findings

R(W_n, S_n) ≤ 5n/2 - 1 for even n ≥ 6

R(W_n, S_n) ≥ 5n/2 - 2 for even n ≥ 6

R(W_n, S_n) = 5n/2 - 2 or 5n/2 - 1 for even n ≥ 6

Abstract

For two graphs $G_1$ and $G_2$ the Ramsey number $R(G_1,G_2)$ is the smallest integer $N$, such that for any graph on $N$ vertices either $G$ contains $G_1$ or $\overline{G}$ contains $G_2$. Let $S_n$ be a star of order $n$ and $W_m$ be a wheel of order $m+1$. In this paper, it is shown that $R(W_n,S_n)\leq{5n/2-1}$, where $n\geq{6}$ is even. It was proven a theorem which implies that $R(W_n,S_n)\geq{5n/2-2}$, where $n\geq{6}$ is even. Therefore we conclude that $R(W_n,S_n)=5n/2-2$ or $5n/2-1$, for $n\geq{6}$ and even.