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arXiv · 2409.10129

Generalized Turán problem for a path and a clique

Xiaona Fang·Xiutao Zhu·Yaojun Chen
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Abstract

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.

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Xiaona Fang, Xiutao Zhu, Yaojun Chen. 2024-09-16. Generalized Turán problem for a path and a clique. https://arxiv.org/abs/2409.10129

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