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Exact generalized Turán number for $K_3$ versus suspension of $P_4$

Sayan Mukherjee

TL;DR

This work resolves the exact value of the generalized Turán number $ ext{ex}(n,K_3,{P}_4)$ for all $n\ge 4$, proving that for $n\ge 8$ the maximum number of triangles is $\big\lfloor n^2/8\big\rfloor$. The authors combine triangle-block analysis and Mantel-type bounds with progressive induction on $n$, using computer-assisted base cases to anchor the inductive step. The main result is that $\text{ex}(n,K_3,\widehat{P}_4)=\left\lfloor n^2/8\right\rfloor$ for all $n\ge 8$, with exact small-$n$ values computed via brute force, and they characterize the extremal structures, including a unique (for large $n$) configuration composed of $K_4$-blocks. The findings close the previously known gap and provide insight into the extremal behavior of triangles in graphs free of the suspension of $P_4$, with implications for related generalized Turán problems and block-decomposition techniques.

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.

Exact generalized Turán number for $K_3$ versus suspension of $P_4$

TL;DR

This work resolves the exact value of the generalized Turán number for all , proving that for the maximum number of triangles is . The authors combine triangle-block analysis and Mantel-type bounds with progressive induction on , using computer-assisted base cases to anchor the inductive step. The main result is that for all , with exact small- values computed via brute force, and they characterize the extremal structures, including a unique (for large ) configuration composed of -blocks. The findings close the previously known gap and provide insight into the extremal behavior of triangles in graphs free of the suspension of , with implications for related generalized Turán problems and block-decomposition techniques.

Abstract

Let denote the path graph on vertices. The suspension of , denoted by , is the graph obtained via adding an extra vertex and joining it to all four vertices of . In this note, we demonstrate that for , the maximum number of triangles in any -vertex graph not containing is . Our method uses simple induction along with computer programming to prove a base case of the induction hypothesis.
Paper Structure (9 sections, 4 theorems, 13 equations, 3 figures)

This paper contains 9 sections, 4 theorems, 13 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Upper bounds
  4. The cases $5\le n\le 8$: brute force
  5. The cases $9\le n \le 11$: identifying $K_4$
  6. The case $n\ge 12$: identifying $K_4$
  7. Case 1: $\sum_{i=0}^4x_i=n-5$.
  8. Case 2: $\sum_{i=0}^4x_i=n-4$.
  9. Concluding Remarks: Uniqueness

Key Result

Theorem 1.1

For $n\ge 8$, $\text{ex}(n, K_3, \widehat{P}_4) = \left\lfloor n^2/8\right\rfloor$. For $n=4,5,6,7$ the values of $\text{ex}(n, K_3,\widehat{P}_4)$ are $4,4,5,8$ respectively.

Figures (3)

  • Figure 1.1: Graphs on $4,5,6,7$ vertices and $4,4,5,8$ triangles, respectively.
  • Figure 2.1: (left): third triangle on $ax_1$, (right): third triangle on $ab$
  • Figure 4.1: A $16$-vertex graph with $32$ triangles consisting of only $K_4$-blocks.

Theorems & Definitions (10)

  • Theorem 1.1
  • Definition 2.1: Triangle-connectivity
  • Definition 2.2: Triangle block
  • Lemma 2.3: suspensionFree2023, Claim 5.3
  • proof
  • Lemma 2.4: hatP3Free-gerbner2022, Section 2
  • proof
  • Theorem 3.1
  • proof
  • proof : Proof of Theorem \ref{['thm:mainthm']} for $n\ge 12$.