On Ramsey number of $K_{2,n}$ versus even cycles
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914538874470400 |
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| author | Dewan, Abisek Gupta, Sayan Mishra, Rajiv |
| author_facet | Dewan, Abisek Gupta, Sayan Mishra, Rajiv |
| contents | For graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the smallest integer $N$ such that every graph $Γ$ on $N$ vertices contains $G$ or its complement $\overlineΓ$ contains $H$ as a subgraph. In graph Ramsey theory, the star-cycle Ramsey number is well-studied throughout the years. Whereas the Ramsey number of $K_{2,n}$ versus cycle is challenging to determine due to increased structural complexity. In this article, we have obtained an exact value of the Ramsey number $R(K_{2,n}, C_{m})$ for even $m\in [n, 2n-4008]$ and $n\geq 4516$. In particular, we show that $$R(K_{1,n}, C_{m})= R(K_{2,n}, C_{m})$$ for all even $m\in [n, 2n-4008]$ and $n\geq 4516$. This leads to an interesting question: For fixed $t$, does there exist $n_0(t)\in \mathbb{N}$ such that $R(K_{1,n}, C_m)=R(K_{t,n}, C_m)$ for all $n \geq n_0(t)$ and for a given range of even $m$? |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_02086 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Ramsey number of $K_{2,n}$ versus even cycles Dewan, Abisek Gupta, Sayan Mishra, Rajiv Combinatorics 05D10, 05C55 (Primary), 05C35 (Secondary) For graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the smallest integer $N$ such that every graph $Γ$ on $N$ vertices contains $G$ or its complement $\overlineΓ$ contains $H$ as a subgraph. In graph Ramsey theory, the star-cycle Ramsey number is well-studied throughout the years. Whereas the Ramsey number of $K_{2,n}$ versus cycle is challenging to determine due to increased structural complexity. In this article, we have obtained an exact value of the Ramsey number $R(K_{2,n}, C_{m})$ for even $m\in [n, 2n-4008]$ and $n\geq 4516$. In particular, we show that $$R(K_{1,n}, C_{m})= R(K_{2,n}, C_{m})$$ for all even $m\in [n, 2n-4008]$ and $n\geq 4516$. This leads to an interesting question: For fixed $t$, does there exist $n_0(t)\in \mathbb{N}$ such that $R(K_{1,n}, C_m)=R(K_{t,n}, C_m)$ for all $n \geq n_0(t)$ and for a given range of even $m$? |
| title | On Ramsey number of $K_{2,n}$ versus even cycles |
| topic | Combinatorics 05D10, 05C55 (Primary), 05C35 (Secondary) |
| url | https://arxiv.org/abs/2604.02086 |