Generalized Turán problem for a path and a clique
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917776597188608 |
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| author | Fang, Xiaona Zhu, Xiutao Chen, Yaojun |
| author_facet | Fang, Xiaona Zhu, Xiutao Chen, Yaojun |
| contents | Let $\mathcal{H}$ be a family of graphs. The generalized Turán number $ex(n, K_r, \mathcal{H})$ is the maximum number of copies of the clique $K_r$ in any $n$-vertex $\mathcal{H}$-free graph. In this paper, we determine the value of $ex(n, K_r, \{P_k, K_m \} )$ for sufficiently large $n$ with an exceptional case, and characterize all corresponding extremal graphs, which generalizes and strengthens the results of Katona and Xiao [EJC, 2024] on $ex(n, K_2, \{P_k, K_m \} )$. For the exceptional case, we obtain a tight upper bound for $ex(n, K_r, \{P_k, K_m \} )$ that confirms a conjecture on $ex(n, K_2, \{P_k, K_m \} )$ posed by Katona and Xiao. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_10129 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalized Turán problem for a path and a clique Fang, Xiaona Zhu, Xiutao Chen, Yaojun Combinatorics Let $\mathcal{H}$ be a family of graphs. The generalized Turán number $ex(n, K_r, \mathcal{H})$ is the maximum number of copies of the clique $K_r$ in any $n$-vertex $\mathcal{H}$-free graph. In this paper, we determine the value of $ex(n, K_r, \{P_k, K_m \} )$ for sufficiently large $n$ with an exceptional case, and characterize all corresponding extremal graphs, which generalizes and strengthens the results of Katona and Xiao [EJC, 2024] on $ex(n, K_2, \{P_k, K_m \} )$. For the exceptional case, we obtain a tight upper bound for $ex(n, K_r, \{P_k, K_m \} )$ that confirms a conjecture on $ex(n, K_2, \{P_k, K_m \} )$ posed by Katona and Xiao. |
| title | Generalized Turán problem for a path and a clique |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2409.10129 |