Turán type problems for a fixed graph and a linear forest
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| Format: | Preprint |
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2025
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| _version_ | 1866915390701961216 |
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| author | Zhang, Haixiang Zhao, Xiamiao Lu, Mei |
| author_facet | Zhang, Haixiang Zhao, Xiamiao Lu, Mei |
| contents | Let $\mathscr{F}$ be a family of graphs. A graph $G$ is $\mathscr{F}$-free if $G$ does not contain any $F\in \mathscr{F}$ as a subgraph. The Turán number, denoted by $ex(n, \mathscr{F})$, is the maximum number of edges in an $n$-vertex $\mathscr{F}$-free graph. Let $F $ be a fixed graph with $ χ(F) \geq 3 $. A forest $H$ is called a linear forest if all components of $H$ are paths. In this paper, we determined the exact value of $ex(n, \{H, F\}) $ for a fixed graph $F$ with $χ(F)\geq 3$ and a linear forest $H$ with at least $2$ components and each component with size at least $3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_11034 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Turán type problems for a fixed graph and a linear forest Zhang, Haixiang Zhao, Xiamiao Lu, Mei Combinatorics Let $\mathscr{F}$ be a family of graphs. A graph $G$ is $\mathscr{F}$-free if $G$ does not contain any $F\in \mathscr{F}$ as a subgraph. The Turán number, denoted by $ex(n, \mathscr{F})$, is the maximum number of edges in an $n$-vertex $\mathscr{F}$-free graph. Let $F $ be a fixed graph with $ χ(F) \geq 3 $. A forest $H$ is called a linear forest if all components of $H$ are paths. In this paper, we determined the exact value of $ex(n, \{H, F\}) $ for a fixed graph $F$ with $χ(F)\geq 3$ and a linear forest $H$ with at least $2$ components and each component with size at least $3$. |
| title | Turán type problems for a fixed graph and a linear forest |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2507.11034 |