Turán problems for suspension of a balanced tree
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866916645833801728 |
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| author | Zhu, Xiutao Wang, Xiaolin Zhang, Yanbo Zhang, Fangfang |
| author_facet | Zhu, Xiutao Wang, Xiaolin Zhang, Yanbo Zhang, Fangfang |
| contents | The Turán number $\ex(n,H)$ is the maximum number of edges that an $n$-vertex $H$-free graph can have. The suspension $\widehat{H}$ is obtained from $H$ by adding a new vertex which is adjacent to all vertices of $H$ and a tree is balanced if the sizes of its two color classes differ at most $1$. In this paper, we obtain a sharp bound of $\ex(n,\widehat{T})$ when $n\ge 4(4k)^6$ based on the Erdős-Sós conjecture. We also show the bound is sharp for infinitely many $n$ and characterize all extremal graphs. In particular, if $T$ satisfies some conditions such as $T$ contains a matching covering all vertices in one color class, then the bound is sharp for all $n$. This is a new class of graphs whose decomposition family does not contain a linear forest but we still can determine its Turán number. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_05166 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Turán problems for suspension of a balanced tree Zhu, Xiutao Wang, Xiaolin Zhang, Yanbo Zhang, Fangfang Combinatorics The Turán number $\ex(n,H)$ is the maximum number of edges that an $n$-vertex $H$-free graph can have. The suspension $\widehat{H}$ is obtained from $H$ by adding a new vertex which is adjacent to all vertices of $H$ and a tree is balanced if the sizes of its two color classes differ at most $1$. In this paper, we obtain a sharp bound of $\ex(n,\widehat{T})$ when $n\ge 4(4k)^6$ based on the Erdős-Sós conjecture. We also show the bound is sharp for infinitely many $n$ and characterize all extremal graphs. In particular, if $T$ satisfies some conditions such as $T$ contains a matching covering all vertices in one color class, then the bound is sharp for all $n$. This is a new class of graphs whose decomposition family does not contain a linear forest but we still can determine its Turán number. |
| title | Turán problems for suspension of a balanced tree |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2503.05166 |