Graph removal lemma

In graph theory, the graph removal lemma states that when a graph contains few copies of a given subgraph, then all of the copies can be eliminated by removing a small number of edges. The special case in which the subgraph is a triangle is known as the triangle removal lemma.

The graph removal lemma can be used to prove Roth's theorem on 3-term arithmetic progressions, and a generalization of it, the hypergraph removal lemma, can be used to prove Szemerédi's theorem. It also has applications to property testing.

Formulation

Let $H$ be a graph with $h$ vertices. The graph removal lemma states that for any $\epsilon >0$, there exists a constant $\delta =\delta (\epsilon ,H)>0$ such that for any $n$-vertex graph $G$ with fewer than $\delta n^{h}$ subgraphs isomorphic to $H$, it is possible to eliminate all copies of $H$ by removing at most $\epsilon n^{2}$ edges from $G$.

An alternative way to state this is to say that for any $n$-vertex graph $G$ with $o(n^{h})$ subgraphs isomorphic to $H$, it is possible to eliminate all copies of $H$ by removing $o(n^{2})$ edges from $G$. Here, the $o$ indicates the use of little o notation.

In the case when $H$ is a triangle, resulting lemma is called triangle removal lemma.

History

The original motivation for the study of triangle removal lemma was Ruzsa–Szemerédi problem. Initial formulation due to Ruzsa and Szemerédi from 1978 was slightly weaker than the triangle removal lemma used nowadays and can be roughly stated as follows: every Locally linear graph on $n$ vertices contains $o(n^{2})$ edges. This statement can be quickly deduced from a modern triangle removal lemma. Ruzsa and Szemerédi provided also an alternative proof of Roth's theorem on arithmetic progressions as a simple corollary.

In 1986 during their work on generalizations of Ruzsa–Szemerédi problem to arbitrary $r$-uniform graphs, Erdős, Frankl, and Rödl provided statement for general graphs very close to the modern graph removal lemma: if graph $H_{2}$ is a homomorphic image of $H_{2}$, then any $H_{1}$-free graph $G$ on $n$ vertices can be made $H_{2}$-free by removing $o(n^{2})$ edges.

The modern formulation of graph removal lemma was first stated by Füredi in 1994. The proof generalized earlier approaches by Ruzsa and Szemerédi and Erdős, Frankl, and Rödl, also utilizing Szemerédi regularity lemma.

Graph counting lemma

A key component of the proof of graph removal lemma is the graph counting lemma about counting subgraphs in systems of regular pairs. Graph counting lemma is also very useful on its own: according to Füredi, it is used "in most applications of regularity lemma".

Heuristic argument

Let $H$ be a graph on $h$ vertices an let $X_{1},X_{2},\ldots ,X_{h}$ be pairwise $\epsilon$-regular pairs (in the sense of regularity lemma) an let $d_{ij}$ be the density between sets $X_{i}$ and $X_{j}$. Intuitively, regular pair $(X,Y)$ with density $d$ should behave like a random Erdős–Rényi-like graph, where every pair $(x,y)\in (X\times Y)$ is selected to be an edge independently with probability $d$. This suggests that the number of copies of $H$ on vertices $x_{1},x_{2},\ldots ,x_{h}$ such that $x_{i}\in X_{i}$ should be close to the expected number from Erdős–Rényi model: $$\prod _{ij\in E(H)}d_{ij}\prod _{i\in V(H)}|X_{i}|$$ where $E(H)$ and $V(H)$ are the edge set and the vertex set of $H$.

Proof

Proof of the triangle removal lemma

To prove the triangle removal lemma, consider an $\epsilon /4$-regular partition $V_{1}\cup \cdots \cup V_{M}$ of the vertex set of $G$. This exists by the Szemerédi regularity lemma. The idea is to remove all edges between irregular pairs, low-density pairs, and small parts, and prove that if at least one triangle still remains, then many triangles remain. Specifically, remove all edges between parts $V_{i}$ and $V_{j}$ if

• $(V_{i},V_{j})$ is not $\epsilon /4$-regular
• the density $d(V_{i},V_{j})$ is less than $\epsilon /2$, or
• either $V_{i}$ or $V_{j}$ has at most $(\epsilon /4M)n$ vertices.

This procedure removes at most $\epsilon n^{2}$ edges. If there exists a triangle with vertices in $V_{i},V_{j},V_{k}$ after these edges are removed, then the triangle counting lemma tells us there are at least $$\left(1-{\frac {\epsilon }{2}}\right)\left({\frac {\epsilon }{4}}\right)^{3}\left({\frac {\epsilon }{4M}}\right)^{3}\cdot n^{3}$$ triples in $V_{i}\times V_{j}\times V_{k}$ which form a triangle. Thus, we may take $$\delta <{\frac {1}{6}}\left(1-{\frac {\epsilon }{2}}\right)\left({\frac {\epsilon }{4}}\right)^{3}\left({\frac {\epsilon }{4M}}\right)^{3}.$$

Proof of the graph removal lemma

The triangle removal lemma was extended to general subgraphs by Erdős, Frankl, and Rödl in 1986. The proof is analogous to the proof of the triangle removal lemma, and uses a generalization of the triangle counting lemma, the graph counting lemma.

Induced Graph Removal Lemma

A natural generalization of the Graph Removal Lemma is to consider induced subgraphs. In property testing it is often useful to consider how far a graph is from being induced H-free. A graph $G$ is considered to contain an induced subgraph $H$ if there is an injective map $f:V(H)\rightarrow V(G)$ such that $(f(u),f(v))$ is an edge of $G$ if and only if $(u,v)$ is an edge of $H$. Notice that non-edges are considered as well. $G$ is induced $H$-free if there is no induced subgraph $G$. We define $G$ as $\epsilon$-far from being induced $H$-free if we cannot add or delete $\epsilon n^{2}$ edges to make $G$ induced $H$-free.

Formulation

A version of the Graph Removal for induced subgraphs was proved by Alon, Fischer, Krivelevich, and Szegedy in 2000. It states that for any graph $H$ with $h$ vertices and $\epsilon >0$, there exists a constant $\delta >0$ such that if an $n$-vertex graph $G$ has fewer than $\delta n^{h}$ induced subgraphs isomorphic to $H$, then it is possible to eliminate all induced copies of $H$ by adding or removing fewer than $\epsilon n^{2}$ edges. Notice that our previous "cleaning" process, where we remove all edges between irregular pairs, low-density pairs, and small parts, does not involve adding edges. Thus, the Regularity Lemma is insufficient to prove Induced Graph Removal Lemma. The proof takes advantage of the strong regularity lemma.

Proof

Strong Regularity Lemma

The strong regularity lemma is a strengthened version of Szemerédi's Regularity Lemma. For any infinite sequence of constants $\epsilon _{0}\geq \epsilon _{1}\geq ...>0$, there exists an integer $M$ such that for any graph $G$, we can obtain two (equitable) partitions ${\mathcal {P}}$ and ${\mathcal {Q}}$ such that the following properties are satisfied:

• ${\mathcal {Q}}$ refines ${\mathcal {P}}$, that is every part of ${\mathcal {P}}$ is the union of some collection of parts in ${\mathcal {Q}}$.
• ${\mathcal {P}}$ is $\epsilon _{0}$-regular and ${\mathcal {Q}}$ is $\epsilon _{|{\mathcal {P}}|}$-regular.
• $q({\mathcal {Q}})<q({\mathcal {P}})+\epsilon _{0}$
• $|{\mathcal {Q}}|\leq M$

The function $q$ is defined to be the energy function defined in Szemerédi regularity lemma. Essentially, we can find a pair of partitions ${\mathcal {P}},{\mathcal {Q}}$ where ${\mathcal {Q}}$ is extremely regular compared to ${\mathcal {P}}$, and at the same time ${\mathcal {P}},{\mathcal {Q}}$ are close to each other. (This property is captured in the third condition)

Corollary of the Strong Regularity Lemma

The following corollary of the strong regularity lemma is used in the proof of the Induced Regularity Lemma. For any infinite sequence of constants $\epsilon _{0}\geq \epsilon _{1}\geq ...>0$, there exists $\delta >0$ such that there exists a partition ${\mathcal {P}}={V_{1},...,V_{k}}$ and subsets $W_{i}\subset V_{i}$ for each $i$ where the following properties are satisfied:

• $|W_{i}|>\delta n$
• $(W_{i},W_{j})$ is $\epsilon _{|{\mathcal {P}}|}$-regular for each pair $i,j$
• $|d(W_{i},W_{j})-d(V_{i},V_{j})|\leq \epsilon _{0}$ for all but $\epsilon _{0}|{\mathcal {P}}|^{2}$ pairs $i,j$

The main idea of the proof of this corollary is to start with two partitions ${\mathcal {P}}$ and ${\mathcal {Q}}$ that satisfy the Strong Regularity Lemma where $q({\mathcal {Q}})<q({\mathcal {P}})+\epsilon _{0}^{3}/8$. Then for each part $V_{i}\in {\mathcal {P}}$, we uniformly at random choose some part $W_{i}\subset V_{i}$ that is a part in ${\mathcal {Q}}$. The expected number of irregular pairs $(W_{i},W_{j})$ is less than 1. Thus, there exists some collection of $W_{i}$ such that every pair is $\epsilon _{|{\mathcal {P}}|}$-regular!

The important aspect of this corollary is that every pair of $W_{i},W_{j}$ are $\epsilon _{|{\mathcal {P}}|}$-regular! This allows us to consider edges and non-edges when we perform our cleaning argument.

Proof of Sketch of the Induced Regularity Lemma

With these results, we are able to prove the Induced Regularity Lemma. Take any graph $G$ with $n$ vertices that has less than $\delta n^{v(H)}$ copies of $H$. The idea is to start with a collection of vertex sets $W_{i}$ which satisfy the conditions of the Corollary of the Strong Regularity Lemma. We then can perform a "cleaning" process where we remove all edges between pairs of parts $(W_{i},W_{j})$ with low density, and we can add all edges between pairs of parts $(W_{i},W_{j})$ with high density. We choose the density requirements such that we added/deleted at most $\epsilon n^{2}$ edges. Suppose the new graph has a copy of $H$. Suppose the vertex $v_{i}\in v(H)$ is embedded in $W_{f(i)}$. Then if there is an edge connecting $v_{i},v_{j}$ in $H$, then $W_{i},W_{j}$ does not have low density. (Edges between $W_{i},W_{j}$ were not removed in the cleaning process) Similarly, if there is not an edge connecting $v_{i},v_{j}$ in $H$, then $W_{i},W_{j}$ does not have high density. (Edges between $W_{i},W_{j}$ were not added in the cleaning process)

Thus, by a similar counting argument to the proof of the triangle counting lemma, that is the graph counting lemma, we can show that $G$ has more than $\delta n^{v(H)}$ copies of $H$.

Generalizations

The graph removal lemma was later extended to directed graphs and to hypergraphs. An alternative proof, providing stronger bounds on the number of edges that need to be removed as a function of the number of copies of the subgraph, was published by Jacob Fox in 2011.

Applications

• Ruzsa and Szemerédi formulated the triangle removal lemma to provide subquadratic upper bounds on the Ruzsa–Szemerédi problem on the size of graphs in which every edge belongs to a unique triangle. This leads to a proof of Roth's theorem.
• The triangle removal lemma can also be used to prove the corners theorem, which states that any subset of $[n]^{2}$ which contains no axis-aligned isosceles right triangle has size $o(n^{2})$.
• The hypergraph removal lemma can be used to prove Szemerédi's theorem on the existence of long arithmetic progressions in dense subsets of integers.
• The graph removal lemma has applications for property testing, because it implies that for every graph, either the graph is near an $H$-free graph, or random sampling will easily find a copy of $H$ in the graph. One result is that for any fixed $\epsilon >0$, there is a constant time algorithm that returns with high probability whether or not a given $n$-vertex graph $G$ is $\epsilon$-far from being $H$-free. Here, $\epsilon$-far from being $H$-free means that at least $\epsilon n^{2}$ edges must be removed to eliminate all copies of $H$ in $G$.
• The induced graph removal lemma was formulated by Alon, Fischer, Krivelevich, and Szegedy to characterize testable graph properties.