Diagonal Ramsey numbers for wheels

TL;DR

This paper improves bounds on the diagonal Ramsey numbers for wheel graphs, providing tighter estimates for even and odd cases and extending to multi-color scenarios.

Contribution

It refines existing bounds on $ ext{R}(W_n,W_n)$ and introduces recursive bounds for multi-colored Ramsey numbers involving wheel graphs.

Findings

Established new bounds: $3n-2 \,\leq\, \mathrm{R}(W_n,W_n) \leq 6n-6$ for even $n\geq 8$

Established new bounds: $2n \leq \mathrm{R}(W_n,W_n) \leq \frac{9n-7}{2}$ for odd $n\geq 7$

Provided recursive bounds for the $k$-colored Ramsey number for $W_n$.

Abstract

The Ramsey number $\mathrm{R}(G_1,G_2)$ is the smallest integer $N$ such that any red-blue coloring of the edges of the complete graph $K_N$ contains either a red copy of $G_1$ or a blue copy of $G_2$. In 2022, the third author and others gave lower and upper bounds of the Ramsey number $\mathrm{R}(W_n,W_n)$, where $W_n$ is the wheel graph with $n$ vertices. In this paper, we improve their bounds by showing that $3n-2\leq \mathrm{R}(W_n,W_n)\leq 6n-6$ for even $n\geq 8$ and $2n\leq \mathrm{R}(W_n,W_n)\leq \frac{9n-7}{2}$ for odd $n\geq 7$. Furthermore, we give recursive bounds for the $k$-colored Ramsey number for $W_n$.