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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2506.01067 |
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| _version_ | 1866910979906863104 |
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| author | Reed, Bruce Yuditsky, Yelena |
| author_facet | Reed, Bruce Yuditsky, Yelena |
| contents | We prove that for every tree $T$ which is not an edge, for almost every graph $G$ which does not contain $T$ as an induced subgraph, $V(G)$ has a partition into $α(T)-1$ parts certifying this fact. Each part induces a graph which is $P_4$-free and has further properties which depend on $T$. As a consequence we obtain good bounds (often tight up to a constant factor) on the number of $T$-free graphs and show in a follow-up paper~\cite{RY} that almost every $T$-free graph $G$ has chromatic number equal to the size of its largest clique. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_01067 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Typical $T$-free graphs Reed, Bruce Yuditsky, Yelena Combinatorics 05C75, 05C80, 05C30 G.2 We prove that for every tree $T$ which is not an edge, for almost every graph $G$ which does not contain $T$ as an induced subgraph, $V(G)$ has a partition into $α(T)-1$ parts certifying this fact. Each part induces a graph which is $P_4$-free and has further properties which depend on $T$. As a consequence we obtain good bounds (often tight up to a constant factor) on the number of $T$-free graphs and show in a follow-up paper~\cite{RY} that almost every $T$-free graph $G$ has chromatic number equal to the size of its largest clique. |
| title | Typical $T$-free graphs |
| topic | Combinatorics 05C75, 05C80, 05C30 G.2 |
| url | https://arxiv.org/abs/2506.01067 |