The maximum number of triangles in graphs without the square of a path

TL;DR

This paper determines the maximum number of triangles in large graphs that do not contain the square of a path with six vertices, extending previous results and characterizing extremal graphs.

Contribution

It provides the exact Turán number for triangles in graphs avoiding the square of a six-vertex path and characterizes all extremal graphs for sufficiently large n.

Findings

Exact value of ex(n, K_3, P_6^2) for n ≥ 11

Characterization of all extremal graphs avoiding P_6^2

Extends previous results for smaller path squares

Abstract

The generalized Tur\'an number for $H$ of $G$, denoted by $\ex(n,H,G)$, is the maximum number of copies of $H$ in an $n$-vertex $G$-free graph. When $H$ is an edge, $\ex(n,H,G)$ is the classical Tur\'an number $\ex(n,G)$. Let $P_k$ be the path with $k$ vertices. The square of $P_k$, denoted by $P_k^2$, is obtained by joining the pairs of vertices with distance at most two in $P_k$. The Tur\'an number of $P_k^2$, $\ex(n, P_k^2)$, was determined by several researchers. When $k=3$, $P_3^2$ is the triangle and $\ex(n, P_3^2)$ is well-known from Mantel's theorem. When $k=4$, $\ex(n, P_4^2)$ was solved by Dirac in a more general context. When $k=5,6$, the problem was solved by Xiao, Katona, Xiao, and Zamora. For general $k \ge 7$, the problem was solved by Yuan in a more general context. Recently, Mukherjee determined the generalized Tur\'an number $\ex(n, K_3, P_5^2)$. In this paper, we determine the exact value of $\ex(n, K_3, P_6^2)$ and characterize all the extremal graphs for $n \ge 11$.