Ramsey Goodness of paths and unbalanced graphs
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arXiv
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| Format: | Preprint |
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2024
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| author | Botler, Fábio Moreira, Luiz de Souza, João Pedro |
| author_facet | Botler, Fábio Moreira, Luiz de Souza, João Pedro |
| contents | Given graphs $G$ and $H$, we say that $G$ is $H$-$good$ if the Ramsey number $R(G,H)$ equals the trivial lower bound $(|G| - 1)(χ(H) - 1) + σ(H)$, where $χ(H)$ denotes the usual chromatic number of $H$, and $σ(H)$ denotes the minimum size of a color class in a $χ(H)$-coloring of $H$. Pokrovskiy and Sudakov [Ramsey goodness of paths. Journal of Combinatorial Theory, Series B, 122:384-390, 2017.] proved that $P_n$ is $H$-good whenever $n\geq 4|H|$. In this paper, given $\varepsilon>0$, we show that if $H$ satisfy a special unbalance condition, then $P_n$ is $H$-good whenever $n \geq (2 + \varepsilon)|H|$. More specifically, we show that if $m_1,\ldots, m_k$ are such that $\varepsilon\cdot m_i \geq 2m_{i-1}^2$ for $2\leq i\leq k$, and $n \geq (2 + \varepsilon)(m_1 + \cdots + m_k)$, then $P_n$ is $K_{m_1,\ldots,m_k}$-good. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_19942 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Ramsey Goodness of paths and unbalanced graphs Botler, Fábio Moreira, Luiz de Souza, João Pedro Combinatorics Discrete Mathematics Given graphs $G$ and $H$, we say that $G$ is $H$-$good$ if the Ramsey number $R(G,H)$ equals the trivial lower bound $(|G| - 1)(χ(H) - 1) + σ(H)$, where $χ(H)$ denotes the usual chromatic number of $H$, and $σ(H)$ denotes the minimum size of a color class in a $χ(H)$-coloring of $H$. Pokrovskiy and Sudakov [Ramsey goodness of paths. Journal of Combinatorial Theory, Series B, 122:384-390, 2017.] proved that $P_n$ is $H$-good whenever $n\geq 4|H|$. In this paper, given $\varepsilon>0$, we show that if $H$ satisfy a special unbalance condition, then $P_n$ is $H$-good whenever $n \geq (2 + \varepsilon)|H|$. More specifically, we show that if $m_1,\ldots, m_k$ are such that $\varepsilon\cdot m_i \geq 2m_{i-1}^2$ for $2\leq i\leq k$, and $n \geq (2 + \varepsilon)(m_1 + \cdots + m_k)$, then $P_n$ is $K_{m_1,\ldots,m_k}$-good. |
| title | Ramsey Goodness of paths and unbalanced graphs |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2410.19942 |