Exact Values and Bounds for Ramsey Numbers of $C_4$ Versus a Star Graph
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| Format: | Preprint |
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2024
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| _version_ | 1866917779902300160 |
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| author | Boza, Luis |
| author_facet | Boza, Luis |
| contents | 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 $R(C_4,K_{1,2n+1-R(C_4,K_{1,n})}) \geq n$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_12770 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Exact Values and Bounds for Ramsey Numbers of $C_4$ Versus a Star Graph Boza, Luis Combinatorics 05C55 G.2.2 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 $R(C_4,K_{1,2n+1-R(C_4,K_{1,n})}) \geq n$. |
| title | Exact Values and Bounds for Ramsey Numbers of $C_4$ Versus a Star Graph |
| topic | Combinatorics 05C55 G.2.2 |
| url | https://arxiv.org/abs/2409.12770 |