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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2512.04402 |
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| _version_ | 1866918231087775744 |
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| author | Luo, Zhidan Peng, Yuejian |
| author_facet | Luo, Zhidan Peng, Yuejian |
| contents | For given graphs $G_{1}, G_{2}$ and $G$, let $G\rightarrow (G_{1}, G_{2})$ denote that each red-blue-coloring of $E(G)$ yields a red copy of $G_{1}$ or a blue copy of $G_{2}$. Arag{ã}o, Marciano and Mendon{\c c}a [L. Arag{ã}o, J. Pedro Marciano and W. Mendon{\c c}a, Degree conditions for Ramsey goodness of paths, {\it European Journal of Combinatorics}, {\bf 124} (2025), 104082] proved the following. Let $G$ be a graph on $N\geq (n- 1)(m- 1)+ 1$ vertices. If $δ(G)\geq N- \lceil n/2\rceil$, then $G\rightarrow (P_{n}, K_{m})$, where $P_{n}$ is a tree on $n$ vertices. In this note, we generalize $P_{n}$ to any tree $T_{n}$ with $n$ vertices, and improve the lower bound of $δ(G)$. We further improve the lower bound when $T_{n}\neq K_{1, n- 1}$, which partially confirms their conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_04402 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A note on degree conditions for Ramsey goodness of trees Luo, Zhidan Peng, Yuejian Combinatorics For given graphs $G_{1}, G_{2}$ and $G$, let $G\rightarrow (G_{1}, G_{2})$ denote that each red-blue-coloring of $E(G)$ yields a red copy of $G_{1}$ or a blue copy of $G_{2}$. Arag{ã}o, Marciano and Mendon{\c c}a [L. Arag{ã}o, J. Pedro Marciano and W. Mendon{\c c}a, Degree conditions for Ramsey goodness of paths, {\it European Journal of Combinatorics}, {\bf 124} (2025), 104082] proved the following. Let $G$ be a graph on $N\geq (n- 1)(m- 1)+ 1$ vertices. If $δ(G)\geq N- \lceil n/2\rceil$, then $G\rightarrow (P_{n}, K_{m})$, where $P_{n}$ is a tree on $n$ vertices. In this note, we generalize $P_{n}$ to any tree $T_{n}$ with $n$ vertices, and improve the lower bound of $δ(G)$. We further improve the lower bound when $T_{n}\neq K_{1, n- 1}$, which partially confirms their conjecture. |
| title | A note on degree conditions for Ramsey goodness of trees |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2512.04402 |