[论文解读] The maximum number of triangles in graphs without the square of a path

Yichen Wang, Győri, Ervin|arXiv (Cornell University)|Jan 14, 2026

Limits and Structures in Graph Theory被引用 0

一句话总结

The paper determines ex(n, K3, P6^2) exactly for n ≥ 11 and identifies the extremal graphs achieving this bound.

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 $n \ge 11$.

研究动机与目标

• Motivate generalized Turán numbers and the square of a path in graph theory.
• Determine the exact value of ex(n, K3, P6^2) for large n and characterize extremal graphs.
• Extend understanding of triangle counts in grids-like triangle graphs and related structures.

提出的方法

• Utilize discharging methods to relate triangle counts to edge counts in P6^2-free graphs.
• Classify blocks of G into four types (K5−-block, K4-block, TP2-block, suspension-block) and analyze their contributions.
• Color edges by blocks to separate blue (triangle-free) versus red components, deriving upper bounds.
• Apply propositions that describe near-Turán structure when the base bipartite part is near maximal.
• Derive upper bounds t(G) ≤ e(G) via discharging and then bound e(G) by known extremal numbers, leading to ex(n, K3, P6^2).
• Provide exact extremal graphs for different n mod 6 and verify optimality.

实验结果

研究问题

• RQ1 What is the exact maximum number of triangles in an n-vertex P6^2-free graph?
• RQ2 What are the extremal graphs attaining ex(n, K3, P6^2) for large n?
• RQ3 How does the structure of blocks (K5−-, K4-, TP2-, suspension-blocks) constrain triangle counts?
• RQ4 How can discharging and near-Turán arguments be combined to bound triangle counts by edge counts?
• RQ5 How do the results depend on n modulo 6?

主要发现

• For n ≥ 11, ex(n, K3, P6^2) = floor(n^2/4) + g(n), where g(n) is given by a mod 6 formula (g(n) values in the paper).
• The maximum is achieved by specific constructions H_n^⌊n/2⌋, F_n^{…}, or related graphs depending on n mod 6, as detailed in the theorem.
• A discharging argument proves t(G) ≤ e(G) for P6^2-free graphs, enabling a tight bound via Mantel’s theorem and Turán-type calculations.
• Blocks in extremal graphs are restricted to four types, and the analysis shows how red triangles are tied to blue-edge counts.
• The authors provide a complete classification of extremal graphs for n ≥ 11 and conjecture the exact value may extend to all n with deeper analysis.