The generalized Turán number for K_3 in graphs without suspensions of a path on five vertices
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918135605493760 |
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| author | Hei, Doudou Hou, Xinmin Ma, Yue |
| author_facet | Hei, Doudou Hou, Xinmin Ma, Yue |
| contents | Given graphs $H$ and $F$, the generalized Turán number $\ex(n, H, F)$ is defined as
the maximum number of copies of $H$ in an $n$-vertex graph that contains no copy of $F$.
The suspension $\widehat{F}$ of a graph $F$ is
obtained by adding a new vertex that is adjacent to every vertex of $F$.
Mubayi and Mukherjee (2023, DM) conjectured that $\ex(n, K_3, \widehat{P_k})=\left\lfloor \frac{k-2}{2}\right\rfloor \cdot \frac{n^2}{8}+o(n^2)$, where $P_k$ is a
path on $k\ge 4$ vertices. Using the triangle removal
lemma, they verified this conjecture for $k=4,5,6$.
Later, Mukherjee (2024, DM) established the exact value $\ex(n, K_3, \widehat{P_4})=\left\lfloor n^2/8\right\rfloor$.
In this paper, using the stability method, we determine the exact value of $\ex(n, K_3, \widehat{P_5})$ by showing that for sufficiently large $n$, $\ex(n,K_3, \widehat{P_5})=\left\lfloor n^2/8\right\rfloor.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_03851 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The generalized Turán number for K_3 in graphs without suspensions of a path on five vertices Hei, Doudou Hou, Xinmin Ma, Yue Combinatorics Given graphs $H$ and $F$, the generalized Turán number $\ex(n, H, F)$ is defined as the maximum number of copies of $H$ in an $n$-vertex graph that contains no copy of $F$. The suspension $\widehat{F}$ of a graph $F$ is obtained by adding a new vertex that is adjacent to every vertex of $F$. Mubayi and Mukherjee (2023, DM) conjectured that $\ex(n, K_3, \widehat{P_k})=\left\lfloor \frac{k-2}{2}\right\rfloor \cdot \frac{n^2}{8}+o(n^2)$, where $P_k$ is a path on $k\ge 4$ vertices. Using the triangle removal lemma, they verified this conjecture for $k=4,5,6$. Later, Mukherjee (2024, DM) established the exact value $\ex(n, K_3, \widehat{P_4})=\left\lfloor n^2/8\right\rfloor$. In this paper, using the stability method, we determine the exact value of $\ex(n, K_3, \widehat{P_5})$ by showing that for sufficiently large $n$, $\ex(n,K_3, \widehat{P_5})=\left\lfloor n^2/8\right\rfloor.$ |
| title | The generalized Turán number for K_3 in graphs without suspensions of a path on five vertices |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2509.03851 |