On Ramsey goodness of $K_{2,n}$ versus cycles
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918488964071424 |
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| author | Dewan, Abisek Gupta, Sayan Mishra, Rajiv |
| author_facet | Dewan, Abisek Gupta, Sayan Mishra, Rajiv |
| contents | A graph $G$ is called $H$-good if $R(G,H)=(|G|-1)(χ(H)-1)+σ(H)$, where $σ(H)$ denotes the size of the smallest color class in a $χ(H)$-coloring of $H$. In Ramsey theory, it is an interesting problem to study whether a graph $G$ is $H$-good or not. In this article, we study the Ramsey goodness of the pair $(K_{2,n},C_m)$, which naturally lies between the classical star-cycle and book-cycle problems. We prove that \begin{equation*}
R(K_{2,n},C_{\{m,m+1\}})=m+1. \end{equation*} for all $m\ge 2n+1$, and consequently establish that \begin{equation*}
R(K_{2,n},C_{m})=m+1. \end{equation*} for all $m\ge 3n+4$. This proves that $C_m$ is $K_{2,n}$-good in this range and improves a particular case of a result on the Ramsey goodness by Pokrovskiy and Sudakov. Further, we provide a construction of a graph that disproves the $C_{m}$-goodness of $K_{2,n}$ for all even $m$ satisfying $n\geq m+2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_06253 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Ramsey goodness of $K_{2,n}$ versus cycles Dewan, Abisek Gupta, Sayan Mishra, Rajiv Combinatorics 05D10, 05C55 (Primary), 05C35 (Secondary) A graph $G$ is called $H$-good if $R(G,H)=(|G|-1)(χ(H)-1)+σ(H)$, where $σ(H)$ denotes the size of the smallest color class in a $χ(H)$-coloring of $H$. In Ramsey theory, it is an interesting problem to study whether a graph $G$ is $H$-good or not. In this article, we study the Ramsey goodness of the pair $(K_{2,n},C_m)$, which naturally lies between the classical star-cycle and book-cycle problems. We prove that \begin{equation*} R(K_{2,n},C_{\{m,m+1\}})=m+1. \end{equation*} for all $m\ge 2n+1$, and consequently establish that \begin{equation*} R(K_{2,n},C_{m})=m+1. \end{equation*} for all $m\ge 3n+4$. This proves that $C_m$ is $K_{2,n}$-good in this range and improves a particular case of a result on the Ramsey goodness by Pokrovskiy and Sudakov. Further, we provide a construction of a graph that disproves the $C_{m}$-goodness of $K_{2,n}$ for all even $m$ satisfying $n\geq m+2$. |
| title | On Ramsey goodness of $K_{2,n}$ versus cycles |
| topic | Combinatorics 05D10, 05C55 (Primary), 05C35 (Secondary) |
| url | https://arxiv.org/abs/2605.06253 |