An exact Ramsey number of large bipartite graphs versus odd wheel
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| Format: | Preprint |
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2025
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| _version_ | 1866915733668102144 |
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| author | Gupta, Sayan Majumder, Kaushik |
| author_facet | Gupta, Sayan Majumder, Kaushik |
| contents | The Ramsey number for the pair of graphs $\mathbb{K}_{1,n}$ (star) versus $W_{m}$ (wheel) has been extensively studied. In contrast, the Ramsey number of $\mathbb{K}_{2,n}$ versus the wheel is not yet explored due to the bit more structural complexity of $\mathbb{K}_{2,n}$ compared to the star. In this article, we have established an exact value of $\mathbb{K}_{2,n}$ versus $W_{m}$ for large $n$ and $m$. In particular, we have proved \begin{equation*} R(\mathbb{K}_{2,n}, W_{m})=3n+4, \end{equation*} whenever $n$ and $m$ are sufficiently large integers satisfying $n\geq4m$ and $m$ is an odd integer. This proves the $W_{m}$-goodness of $\mathbb{K}_{2,n}$. Our proof combines probabilistic methods with an analysis of structural dependencies. As part of the argument, we resolve a structural rigidity question concerning highly dependent neighbourhoods (Lemma 3.12). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_14867 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An exact Ramsey number of large bipartite graphs versus odd wheel Gupta, Sayan Majumder, Kaushik Combinatorics Primary: 05C55, 05D10, 05D40. Secondary: 05C35 The Ramsey number for the pair of graphs $\mathbb{K}_{1,n}$ (star) versus $W_{m}$ (wheel) has been extensively studied. In contrast, the Ramsey number of $\mathbb{K}_{2,n}$ versus the wheel is not yet explored due to the bit more structural complexity of $\mathbb{K}_{2,n}$ compared to the star. In this article, we have established an exact value of $\mathbb{K}_{2,n}$ versus $W_{m}$ for large $n$ and $m$. In particular, we have proved \begin{equation*} R(\mathbb{K}_{2,n}, W_{m})=3n+4, \end{equation*} whenever $n$ and $m$ are sufficiently large integers satisfying $n\geq4m$ and $m$ is an odd integer. This proves the $W_{m}$-goodness of $\mathbb{K}_{2,n}$. Our proof combines probabilistic methods with an analysis of structural dependencies. As part of the argument, we resolve a structural rigidity question concerning highly dependent neighbourhoods (Lemma 3.12). |
| title | An exact Ramsey number of large bipartite graphs versus odd wheel |
| topic | Combinatorics Primary: 05C55, 05D10, 05D40. Secondary: 05C35 |
| url | https://arxiv.org/abs/2511.14867 |