Overview
This research focuses on diagonal Ramsey numbers for wheel graphs, specifically addressing $\mathrm{R}(W_n,W_n)$. The study provides improved lower and upper bounds for this specific Ramsey number for both even and odd values of $n$. Additionally, recursive bounds for the $k$-colored Ramsey number for $W_n$ are presented.
Research Context
The Ramsey number $\mathrm{R}(G_1,G_2)$ is defined as the smallest integer $N$ such that any red-blue coloring of the edges of a complete graph $K_N$ must contain either a red copy of graph $G_1$ or a blue copy of graph $G_2$. A prior study in 2022, co-authored by the third author of this paper, established initial lower and upper bounds for $\mathrm{R}(W_n,W_n)$, where $W_n$ denotes the wheel graph with $n$ vertices. The current research aims to refine these previously established bounds.
Approach
The paper describes an improvement upon existing bounds for the diagonal Ramsey number $\mathrm{R}(W_n,W_n)$. The methodology involves deriving new inequalities that establish tighter lower and upper limits for this specific type of Ramsey number. The work also extended to formulate recursive bounds for the $k$-colored Ramsey number for $W_n$, which involves scenarios with more than two colors.
Findings
- For even $n\geq 8$, the improved bounds for the diagonal Ramsey number $\mathrm{R}(W_n,W_n)$ are found to be $3n-2\leq \mathrm{R}(W_n,W_n)\leq 6n-6$.
- For odd $n\geq 7$, the improved bounds for the diagonal Ramsey number $\mathrm{R}(W_n,W_n)$ are determined to be $2n\leq \mathrm{R}(W_n,W_n)\leq \frac{9n-7}{2}$.
- The research also provides recursive bounds for the $k$-colored Ramsey number for $W_n$.