A note on the number of triangles in graphs without the suspension of a path on four vertices

TL;DR

This paper determines the maximum number of triangles in large graphs that do not contain a specific suspended path structure, confirming a precise asymptotic value for sufficiently large graphs.

Contribution

It proves that for large graphs, the maximum number of triangles avoiding the suspension of P4 is exactly the floor of n^2/8, refining previous asymptotic bounds.

Findings

Exact maximum number of triangles for large graphs is floor(n^2/8).

Confirms previous asymptotic results with precise value.

Provides structural insights into graphs avoiding the suspension of P4.

Abstract

The suspension of the path $P_4$ consists of a $P_4$ and an additional vertex connected to each of the four vertices, and is denoted by $\hat{P_4}$. The largest number of triangles in a $\hat{P_4}$-free $n$-vertex graph is denoted by $ex(n,K_3,\hat{P_4})$. Mubayi and Mukherjee in 2020 showed that $ ex(n,K_3,\hat{P_4})= n^2/8+O(n)$. We show that for sufficiently large $n$, $ex(n,K_3,\hat{P_4})=\lfloor n^2/8\rfloor$.