The maximum number of triangles in $F_k$-free graphs

Xiutao Zhu; Yaojun Chen; Dániel Gerbner; Ervin Győri; Hilal Hama Karim

The maximum number of triangles in $F_k$-free graphs

Combinatorics 2022-08-19 v2

Authors: Xiutao Zhu , Yaojun Chen , Dániel Gerbner , Ervin Győri , Hilal Hama Karim

Abstract

The generalized Tur\'{a}n number $ex(n,K_s,H)$ is the maximum number of complete graph $K_s$ in an $H$ -free graph on $n$ vertices. Let $F_k$ be the friendship graph consisting of $k$ triangles. Erd\H{o}s and S\'os (1976) determined the value of $ex(n,K_3,F_2)$ . Alon and Shikhelman (2016) proved that $ex(n,K_3, F_k)\le (9k-15)(k+1)n.$ In this paper, by using a method developed by Chung and Frankl in hypergraph theory, we determine the exact value of $ex(n,K_3,F_k)$ and the extremal graph for any $F_k$ when $n\ge 4k^3$ .

Keywords

generalized turán number friendship graph triangle extremal graph hypergraph method free graph

Cite

@article{arxiv.2207.10162,
  title  = {The maximum number of triangles in $F_k$-free graphs},
  author = {Xiutao Zhu and Yaojun Chen and Dániel Gerbner and Ervin Győri and Hilal Hama Karim},
  journal= {arXiv preprint arXiv:2207.10162},
  year   = {2022}
}

The maximum number of triangles in $F_k$-free graphs

Abstract

Keywords

Cite

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