[Paper Review] Triangles in graphs without bipartite suspensions

Dhruv Mubayi, Sayan Mukherjee|arXiv (Cornell University)|Apr 24, 2020

Limits and Structures in Graph Theory4 citations

TL;DR

This paper establishes improved upper bounds on the generalized Turán number ex(n, K₃, Ĥ), where Ĥ is the suspension of a graph H, focusing on H being a complete bipartite graph K_{a,b}, an even cycle C_{2k}, or a path P_k. Using the triangle removal lemma and structural analysis of neighborhoods, it proves that ex(n, K₃, K̂_{1,a,b}) = o(n^{3−1/a}) and ex(n, K₃, K̂_{1,2,2}) = o(n^{5/2}), significantly improving prior bounds and narrowing the gap toward a conjectured quadratic bound.

ABSTRACT

Given graphs $T$ and $H$, the generalized Turán number ex$(n,T,H)$ is the maximum number of copies of $T$ in an $n$-vertex graph with no copies of $H$. Alon and Shikhelman, using a result of Erd\H os, determined the asymptotics of ex$(n,K_3,H)$ when the chromatic number of $H$ is greater than 3 and proved several results when $H$ is bipartite. We consider this problem when $H$ has chromatic number 3. Even this special case for the following relatively simple 3-chromatic graphs appears to be challenging. The suspension $\widehat H$ of a graph $H$ is the graph obtained from $H$ by adding a new vertex adjacent to all vertices of $H$. We give new upper and lower bounds on ex$(n,K_3,\widehat{H})$ when $H$ is a path, even cycle, or complete bipartite graph. One of the main tools we use is the triangle removal lemma, but it is unclear if much stronger statements can be proved without using the removal lemma.

Motivation & Objective

• To determine tighter upper bounds for the number of triangles in n-vertex graphs that exclude the suspension of a 3-chromatic graph H, i.e., Ĥ = K₁ ∨ H.
• To address the open problem of whether ex(n, K₃, K̂_{1,2,2}) is O(n²), providing a super-quadratic upper bound and a quadratic lower bound.
• To extend the analysis beyond known results for χ(H) > 3 and χ(H) = 2, focusing on the challenging case χ(H) = 3.
• To explore the role of the triangle removal lemma and neighborhood structure in deriving improved asymptotic bounds for generalized Turán problems.

Proposed method

• Use the inequality ex(n, K₃, Ĥ) ≤ (n/3) · ex(n, H) as a baseline, then refine it using structural constraints on neighborhoods in Ĥ-free graphs.
• Apply the triangle removal lemma to show that graphs with few triangles must have a bounded number of edge-disjoint triangles, enabling decomposition techniques.
• Analyze the codegree and edge types (light/heavy) in triangles to derive inductive bounds on triangle counts via edge deletion.
• Use the Kövari–Sós–Turán theorem for K_{a,b}-free graphs and Bondy–Simonovits bounds for C_{2k}-free graphs to bound ex(n, H) for H = K_{a,b}, C_{2k}, P_k.
• Employ Jensen’s inequality on the sum of neighborhood degrees raised to power (α−1) to bound the total triangle count in terms of ex(n, H) = O(n^α).
• Generalize results via Theorem 6.1: if ex(n, H) = O(n^α) for 1 < α < 2, then ex(n, K₃, Ĥ) = o(n^{1+α}), unifying the analysis across H types.

Experimental results

Research questions

• RQ1 Can the bound ex(n, K₃, K̂_{1,a,b}) = O(n^{3−1/a}) be improved to o(n^{3−1/a}) for fixed a ≥ 1, b ≥ a?
• RQ2 Is ex(n, K₃, K̂_{1,2,2}) = O(n²), or does it grow strictly faster than quadratic, as suggested by the current o(n^{5/2}) upper bound?
• RQ3 What is the true asymptotic growth of ex(n, K₃, K̂_{P_k}) for fixed k ≥ 3, and is the quadratic lower bound Ω(n²) tight?
• RQ4 To what extent can the triangle removal lemma be avoided in proving such generalized Turán bounds, or is it essential for strong results?
• RQ5 Can the bound ex(n, K₃, Ĥ) = o(n^{1+α}) be established uniformly for all H with ex(n, H) = O(n^α), 1 < α < 2?

Key findings

• For fixed 1 ≤ a ≤ b, ex(n, K₃, K̂_{1,a,b}) = o(n^{3−1/a}), improving the prior O(n^{3−1/a}) bound from the Kövari–Sós–Turán theorem.
• For H = C_{2k}, ex(n, K₃, K̂_{C_{2k}}) = o(n^{2+1/k}), improving the classical O(n^{2+1/k}) bound from Bondy–Simonovits.
• For H = P_k with k ≥ 3, ex(n, K₃, K̂_{P_k}) ≥ ⌊(k−1)/2⌋ · n²/8, providing a quadratic lower bound when n is divisible by 4⌊(k−1)/2⌋.
• For k = 3, 4, 5, ex(n, K₃, K̂_{P_k}) = ⌊(k−1)/2⌋ · n²/8 + o(n²), showing the lower bound is asymptotically tight in these cases.
• Theorem 6.1 establishes a general result: if ex(n, H) = O(n^α) for 1 < α < 2, then ex(n, K₃, Ĥ) = o(n^{1+α}), unifying the analysis across H types.
• The paper demonstrates that the triangle removal lemma is essential for achieving strong o(n^{1+α}) bounds, as alternative methods are insufficient.