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

TL;DR

The paper resolves the generalized Turán problem $ ext{ex}(n,K_3,P_6^2)$ for $n\ge 11$ by developing a discharging argument to show $t(G)\le e(G)$ for $P_6^2$-free graphs and a block-decomposition analysis that classifies extremal structures into four block types. By combining these tools, the authors establish the exact bound $ ext{ex}(n,K_3,P_6^2)=t(n,2)+g(n)$ with $t(n,2)=\lfloor n^2/4\rfloor$ and a modularly defined $g(n)$, and provide a complete description of all extremal graphs via the constructions $H_n^i$ and $F_n^{i,j}$ depending on $n\bmod 6$. The approach hinges on a tight relationship between triangle counts and edges, plus a careful block-level analysis that yields near-bipartite blue subgraphs augmented by a controlled number of $K_5^-$-blocks. The results extend prior work on $P_k^2$ and triangle-Turán problems and suggest directions for tackling larger squares of paths and related triangle-grid configurations, with potential applications to TP$_k$-free extremal questions.

Abstract

The generalized Turán 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 Turán 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 Turán 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 Turán 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 $n \ge 11$.

Figures (11)

• Figure 1: The square of $P_k$ denoted by $P_k^2$.
• Figure 2: Graphs $F_{n}^{i,j}$ and $H_{n}^{i}$.
• Figure 3: The Triangular Pyramid graphs: $TP_2$, $TP_3$ and $TP_4$.
• Figure 4: The configurations and extremal structures in Proposition \ref{['prop: almost bipartite with triangle']}.
• Figure 5: The configuration in the proof of Claim \ref{['claim: diamond']}. $3^+$ means the type of $xw$ is at least $3$.

Theorems & Definitions (23)

• Theorem 1.1: mantel1907vraagstuk
• Theorem 1.2: XIAO20221
• Definition 1.3
• Theorem 1.4: XIAO20221
• Theorem 1.5: YUAN2022103548
• Definition 1.6
• Theorem 1.7: XIAO20221
• Theorem 1.8: GHOSH202275
• Theorem 1.9: MUKHERJEE2024113866
• Theorem 1.10: LV2024113682