Turán number of complete bipartite graphs with bounded matching number
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910576252289024 |
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| author | Luo, Huan Zhao, Xiamiao Lu, Mei |
| author_facet | Luo, Huan 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 $F\in \mathcal{F}$ as a subgraph. The Turán number $ex(n, \mathscr{F})$ is the maximum number of edges in an $n$-vertex $\mathscr{F}$-free graph. Let $M_{s}$ be the matching consisting of $ s $ independent edges. Recently, Alon and Frank determined the exact value of $ex(n,\{K_{m},M_{s+1}\})$. Gerbner obtained several results about $ex(n,\{F,M_{s+1}\})$ when $F$ satisfies certain proportions. In this paper, we determine the exact value of $ex(n,\{K_{l,t},M_{s+1}\})$ when $s, n$ are large enough for every $3\leq l\leq t$. When $n$ is large enough, we also show that $ex(n,\{K_{2,2}, M_{s+1}\})=n+{s\choose 2}-\left\lceil\frac{s}{2}\right\rceil$ for $s\ge 12$ and $ex(n,\{K_{2,t},M_{s+1}\})=n+(t-1){s\choose 2}-\left\lceil\frac{s}{2}\right\rceil$ when $t\ge 3$ and $s$ is large enough. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_13994 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Turán number of complete bipartite graphs with bounded matching number Luo, Huan 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 $F\in \mathcal{F}$ as a subgraph. The Turán number $ex(n, \mathscr{F})$ is the maximum number of edges in an $n$-vertex $\mathscr{F}$-free graph. Let $M_{s}$ be the matching consisting of $ s $ independent edges. Recently, Alon and Frank determined the exact value of $ex(n,\{K_{m},M_{s+1}\})$. Gerbner obtained several results about $ex(n,\{F,M_{s+1}\})$ when $F$ satisfies certain proportions. In this paper, we determine the exact value of $ex(n,\{K_{l,t},M_{s+1}\})$ when $s, n$ are large enough for every $3\leq l\leq t$. When $n$ is large enough, we also show that $ex(n,\{K_{2,2}, M_{s+1}\})=n+{s\choose 2}-\left\lceil\frac{s}{2}\right\rceil$ for $s\ge 12$ and $ex(n,\{K_{2,t},M_{s+1}\})=n+(t-1){s\choose 2}-\left\lceil\frac{s}{2}\right\rceil$ when $t\ge 3$ and $s$ is large enough. |
| title | Turán number of complete bipartite graphs with bounded matching number |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2408.13994 |