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Generalized Tur\'an problem for a path and a clique

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arxiv 2409.10129 v1 pith:C66HELRZ submitted 2024-09-16 math.CO

Xiaona Fang , Xiutao Zhu , Yaojun Chen This is my paper · ORCID
classification math.CO
keywords mathcalcasecliqueexceptionalgeneralizedgraphskatonanumber
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abstract

Let $\mathcal{H}$ be a family of graphs. The generalized Tur\'an 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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  1. Tur\'an numbers of cycles plus a general graph

    math.CO 2024-11 conditional novelty 6.0 of 10

    The Turan number ex(n,{C>=k,F}) is determined up to an additive constant for every 2-connected F with p(F) at least floor((k-1)/2)+1; the even-k formula is n times the larger of (k-2)/2 and ex(k-1,F)/(k-2), plus O_k(1).

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