Exact Values and Bounds for Ramsey Numbers of $C_4$ Versus a Star Graph

TL;DR

This paper determines exact values and bounds for Ramsey numbers involving a 4-cycle versus star graphs, advancing understanding of these combinatorial quantities through new calculations and inequalities.

Contribution

It provides exact values for previously unknown Ramsey numbers and establishes new bounds and inequalities for these numbers involving $C_4$ and star graphs.

Findings

Determined 8 unknown Ramsey numbers for $R(C_4,K_{1,n})$ with $n \,\leq\, 37$.

Established bounds for $R(C_4,K_{1,n})$ based on parity and modular conditions.

Proved inequalities relating the growth of Ramsey numbers for different $n$.

Abstract

The 8 unknown values of the Ramsey numbers $R(C_4,K_{1,n})$ for $n \leq 37$ are determined, showing that $R(C_4,K_{1,27}) = 33$ and $R(C_4,K_{1,n}) = n + 7$ for $28 \leq n \leq 33$ or $n = 37$. Additionally, the following results are proven: $\bullet$ If $n$ is even and $\lceil\sqrt{n}\rceil$ is odd, then $R(C_4,K_{1,n}) \leq n + \left\lceil\sqrt{n-\lceil\sqrt{n}\rceil+2}\right\rceil + 1$. $\bullet$ If $m \equiv 2 \,(\text{mod } 6)$ with $m \geq 8$, then $R(C_4,K_{1,m^2+3}) \leq m^2 + m + 4$. $\bullet$ If $R(C_4,K_{1,n}) > R(C_4,K_{1,n-1})$, then $R(C_4,K_{1,2n+1-R(C_4,K_{1,n})}) \geq n$.