Trichotomy and $tK_m$-goodness of sparse graphs
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866914188450856960 |
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| author | Zhang, Yanbo Chen, Yaojun |
| author_facet | Zhang, Yanbo Chen, Yaojun |
| contents | Let $G$ be a connected graph with $n$ vertices and $n+k-2$ edges and $tK_m$ denote the disjoint union of $t$ complete graphs $K_m$. In this paper, by developing a trichotomy for sparse graphs, we show that for given integers $m\ge 2$ and $t\ge 1$, there exists a positive constant $c$ such that if $1\le k\le cn^{\frac{2}{m-1}}$ and $n$ is large, then $G$ is $tK_m$-good, that is, the Ramsey number is \[ r(G, tK_m)=(n-1)(m-1)+t\,. \] In particular, the above equality holds for any positive integers $k$, $m$, and $t$, provided $n$ is large. The case $t=1$ was obtained by Burr, Erdős, Faudree, Rousseau, and Schelp (1980), and the case $k=1$ was established by Luo and Peng (2023). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_04142 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Trichotomy and $tK_m$-goodness of sparse graphs Zhang, Yanbo Chen, Yaojun Combinatorics 05C55, 05D10 Let $G$ be a connected graph with $n$ vertices and $n+k-2$ edges and $tK_m$ denote the disjoint union of $t$ complete graphs $K_m$. In this paper, by developing a trichotomy for sparse graphs, we show that for given integers $m\ge 2$ and $t\ge 1$, there exists a positive constant $c$ such that if $1\le k\le cn^{\frac{2}{m-1}}$ and $n$ is large, then $G$ is $tK_m$-good, that is, the Ramsey number is \[ r(G, tK_m)=(n-1)(m-1)+t\,. \] In particular, the above equality holds for any positive integers $k$, $m$, and $t$, provided $n$ is large. The case $t=1$ was obtained by Burr, Erdős, Faudree, Rousseau, and Schelp (1980), and the case $k=1$ was established by Luo and Peng (2023). |
| title | Trichotomy and $tK_m$-goodness of sparse graphs |
| topic | Combinatorics 05C55, 05D10 |
| url | https://arxiv.org/abs/2505.04142 |