Exact generalized Tur\'an number for $K_3$ versus suspension of $P_4$

TL;DR

This paper determines the maximum number of triangles in large graphs that do not contain the suspension of a 4-vertex path, using induction and computer assistance to establish an exact Turán number.

Contribution

It provides the exact generalized Turán number for triangles versus the suspension of P4, a previously unresolved extremal graph problem.

Findings

Maximum triangles in n-vertex graphs without suspension of P4 is floor(n^2/8) for n ≥ 8.

Uses induction and computer programming to prove the result.

Establishes a new extremal graph theory result for a specific forbidden subgraph.

Abstract

Let $P_4$ denote the path graph on $4$ vertices. The suspension of $P_4$, denoted by $\widehat P_4$, is the graph obtained via adding an extra vertex and joining it to all four vertices of $P_4$. In this note, we demonstrate that for $n\ge 8$, the maximum number of triangles in any $n$-vertex graph not containing $\widehat P_4$ is $\left\lfloor n^2/8\right\rfloor$. Our method uses simple induction along with computer programming to prove a base case of the induction hypothesis.