Counting cliques without generalized theta graphs
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866908848968695808 |
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| author | Gao, Jun Wu, Zhuo Xue, Yisai |
| author_facet | Gao, Jun Wu, Zhuo Xue, Yisai |
| contents | The \textit{generalized Turán number} $\mathrm{ex}(n, T, F)$ is the maximum possible number of copies of $T$ in an $F$-free graph on $n$ vertices for any two graphs $T$ and $F$. For the book graph $B_t$, there is a close connection between $\ex(n,K_3,B_t)$ and the Ruzsa-Szemerédi triangle removal lemma. Motivated by this, in this paper, we study the generalized Turán problem for generalized theta graphs, a natural extension of book graphs. Our main result provides a complete characterization of the magnitude of $\ex(n,K_3,H)$ when $H$ is a generalized theta graph, indicating when it is quadratic, when it is nearly quadratic, and when it is subquadratic. Furthermore, as an application, we obtain the exact value of $\ex(n, K_r, kF)$, where $F$ is an edge-critical generalized theta graph, and $3\le r\le k+1$, extending several recent results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_15289 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Counting cliques without generalized theta graphs Gao, Jun Wu, Zhuo Xue, Yisai Combinatorics The \textit{generalized Turán number} $\mathrm{ex}(n, T, F)$ is the maximum possible number of copies of $T$ in an $F$-free graph on $n$ vertices for any two graphs $T$ and $F$. For the book graph $B_t$, there is a close connection between $\ex(n,K_3,B_t)$ and the Ruzsa-Szemerédi triangle removal lemma. Motivated by this, in this paper, we study the generalized Turán problem for generalized theta graphs, a natural extension of book graphs. Our main result provides a complete characterization of the magnitude of $\ex(n,K_3,H)$ when $H$ is a generalized theta graph, indicating when it is quadratic, when it is nearly quadratic, and when it is subquadratic. Furthermore, as an application, we obtain the exact value of $\ex(n, K_r, kF)$, where $F$ is an edge-critical generalized theta graph, and $3\le r\le k+1$, extending several recent results. |
| title | Counting cliques without generalized theta graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2311.15289 |