[Paper Review] The regularity method for graphs with few 4-cycles

David Conlon, Jacob Fox+2 more|arXiv (Cornell University)|Apr 21, 2020

Limits and Structures in Graph Theory27 references7 citations

TL;DR

This paper develops a sparse graph regularity method tailored for graphs with few 4-cycles, establishing a novel counting lemma for 5-cycles that enables new removal lemmas and applications in extremal and additive combinatorics. The key contribution is a sparse $C_5$-removal lemma in $C_4$-free graphs, showing that $o(n^{5/2})$ five-cycles can be removed by deleting $o(n^{3/2})$ edges, with applications to triangle removal and equations with no nontrivial solutions.

ABSTRACT

We develop a sparse graph regularity method that applies to graphs with few 4-cycles, including new counting and removal lemmas for 5-cycles in such graphs. Some applications include: * Every $n$-vertex graph with no 5-cycle can be made triangle-free by deleting $o(n^{3/2})$ edges. * For $r \geq 3$, every $n$-vertex $r$-graph with girth greater than $5$ has $o(n^{3/2})$ edges. * Every subset of $[n]$ without a nontrivial solution to the equation $x_1 + x_2 + 2x_3 = x_4 + 3x_5$ has size $o(\sqrt{n})$.

Motivation & Objective

• To develop a sparse regularity method applicable to graphs with few 4-cycles, without requiring pseudorandom or random host graphs.
• To establish a new counting lemma for 5-cycles in such graphs, enabling removal lemmas and structural applications.
• To apply the method to extremal graph theory, proving that $C_5$-free graphs can be made triangle-free by removing $o(n^{3/2})$ edges.
• To derive bounds on the size of sets without nontrivial solutions to specific linear equations, using graph-theoretic constructions.

Proposed method

• The method introduces a counting lemma that lower bounds the number of 5-cycles in $C_4$-free graphs using the sparse regularity framework.
• It leverages the absence of 4-cycles to control the number of 5-cycle embeddings and derive structural constraints.
• The approach combines graph regularity with extremal combinatorics, focusing on graphs where the number of 4-cycles is subquadratic.
• A key construction uses vertex sets indexed by $\mathbb{Z}/N\mathbb{Z}$ and edge sets defined by a Sidon set $X$ to generate edge-disjoint 5-cycles.
• The method avoids reliance on pseudorandom or random host graphs, instead using the sparsity and cycle-freeness of the graph as structural assumptions.
• It applies probabilistic and algebraic techniques, including tensor powers of triangles and Markov's inequality, to construct extremal examples.

Experimental results

Research questions

• RQ1 Can a removal lemma for 5-cycles be established in $C_4$-free graphs without assuming pseudorandomness or containment in a random graph?
• RQ2 What is the extremal relationship between the number of 5-cycles and the number of edges needed to remove to make a graph $C_5$-free in $C_4$-free graphs?
• RQ3 Can the regularity method be adapted to sparse graphs with few 4-cycles to yield new results in extremal graph theory and additive combinatorics?
• RQ4 What is the maximum size of a subset of $[n]$ with no nontrivial solutions to $x_1 + x_2 + 2x_3 = x_4 + 3x_5$?
• RQ5 Can the exponent in the bound on the number of $C_5$'s in graphs that cannot be made triangle-free by removing $o(n^{3/2})$ edges be improved?

Key findings

• An $n$-vertex $C_4$-free graph with $o(n^{5/2})$ five-cycles can be made $C_5$-free by removing $o(n^{3/2})$ edges, establishing a sparse $C_5$-removal lemma.
• Every $n$-vertex graph with no $C_5$ can be made triangle-free by deleting $o(n^{3/2})$ edges, extending the triangle removal lemma to sparse settings.
• For $r \geq 3$, every $n$-vertex $r$-uniform hypergraph with girth greater than 5 has $o(n^{3/2})$ edges, a new extremal result.
• There exist $n$-vertex graphs with $o(n^{2.442})$ five-cycles that cannot be made triangle-free by removing $o(n^{3/2})$ edges, showing the exponent is tight up to constant factors.
• A subset of $[n]$ without nontrivial solutions to $x_1 + x_2 + 2x_3 = x_4 + 3x_5$ has size $o(\sqrt{n})$, resolving a problem in additive combinatorics.
• The construction of a $C_5$-free graph with $\Theta(n^{3/2})$ edge-disjoint triangles and $o(n^{2.442})$ five-cycles demonstrates the tightness of the exponent in the $C_5$-counting bound.