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.[1] The special case in which the subgraph is a triangle is known as the triangle removal lemma.[2]

The graph removal lemma can be used to prove Roth's theorem on 3-term arithmetic progressions,[3] and a generalization of it, the hypergraph removal lemma, can be used to prove Szemerédi's theorem.[4] It also has applications to property testing.[5]

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 $G$ .[1]

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.

Proof

Proof of the triangle removal lemma

The graph removal lemma was originally proven for the case that the subgraph is a triangle by Imre Z. Ruzsa and Endre Szemerédi in 1978, using the Szemerédi regularity lemma.[6] A key element of the proof is the triangle counting lemma.

Informally, if vertex subsets $X,Y,Z$ of $G$ are "random-like" with pairwise edge densities $d_{XY},d_{XZ},d_{YZ}$ , then we expect the number of triples $(x,y,z)\in X\times Y\times Z$ such that $x,y,z$ form a triangle in $G$ to be

d_{XY}d_{XZ}d_{YZ}\cdot |X||Y||Z|.

More precisely, suppose vertex subsets $X,Y,Z$ of $G$ are pairwise $\epsilon$ -regular, and suppose the edge densities $d_{XY},d_{XZ},d_{YZ}$ are all at least $2\epsilon$ . Then the number of triples $(x,y,z)\in X\times Y\times Z$ such that $x,y,z$ form a triangle in $G$ is at least

(1-2\epsilon )(d_{XY}-\epsilon )(d_{XZ}-\epsilon )(d_{YZ}-\epsilon )\cdot |X||Y||Z|.

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.[7] 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.

Generalizations

The graph removal lemma was later extended to directed graphs[5] and to hypergraphs.[4] 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.[1]

A version for induced subgraphs was proved by Alon, Fischer, Krivelevich, and Szegedy in 2000.[8] 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. Note that the proof of the graph removal lemma does not easily extend to induced subgraphs because given a regular partition of the vertex set of $G$ , it now becomes unclear whether to add or remove all the edges between irregular pairs. This issue must be remedied using the strong regularity lemma.[8]

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.[3]
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 $o(n^{2})$ .[9]
The hypergraph removal lemma can be used to prove Szemerédi's theorem on the existence of long arithmetic progressions in dense subsets of integers.[4]
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.[5] 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.[10] 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.[8]

References

Fox, Jacob (2011), "A new proof of the graph removal lemma", Annals of Mathematics, Second Series, 174 (1): 561–579, arXiv:1006.1300, doi:10.4007/annals.2011.174.1.17, MR 2811609
Trevisan, Luca (May 13, 2009), "The Triangle Removal Lemma", in theory
Roth, K. F. (1953), "On certain sets of integers", Journal of the London Mathematical Society, 28 (1): 104–109, doi:10.1112/jlms/s1-28.1.104, MR 0051853
Tao, Terence (2006), "A variant of the hypergraph removal lemma", Journal of Combinatorial Theory, Series A, 113 (7): 1257–1280, arXiv:math/0503572, doi:10.1016/j.jcta.2005.11.006, MR 2259060
Alon, Noga; Shapira, Asaf (2004), "Testing subgraphs in directed graphs", Journal of Computer and System Sciences, 69 (3): 353–382, doi:10.1016/j.jcss.2004.04.008, MR 2087940
Ruzsa, I. Z.; Szemerédi, E. (1978), "Triple systems with no six points carrying three triangles", Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. II, Colloq. Math. Soc. János Bolyai, 18, Amsterdam and New York: North-Holland, pp. 939–945, MR 0519318
Erdős, P.; Frankl, P.; Rödl, V. (1986), "The asymptotic number of graphs not containing a fixed subgraph and a problem for hypergraphs having no exponent", Graphs and Combinatorics, 2 (2): 113–121, doi:10.1007/BF01788085, MR 0932119
Alon, Noga; Fischer, Eldar; Krivelevich, Michael; Szegedy, Mario (2000), "Efficient testing of large graphs", Combinatorica, 20 (4): 451–476, doi:10.1007/s004930070001, MR 1804820
Solymosi, J. (2003), "Note on a generalization of Roth's theorem", Discrete and Computational Geometry, Algorithms and Combinatorics, 25: 825–827, doi:10.1007/978-3-642-55566-4_39, ISBN 978-3-642-62442-1, MR 2038505, S2CID 53973423
Alon, Noga; Shapira, Asaf (2008), "A characterization of the (natural) graph properties testable with one-sided error", SIAM Journal on Computing, 37 (6): 1703–1727, doi:10.1137/06064888X, MR 2386211

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[f11-1] Fox, Jacob (2011), "A new proof of the graph removal lemma", Annals of Mathematics, Second Series, 174 (1): 561–579, arXiv:1006.1300, doi:10.4007/annals.2011.174.1.17, MR 2811609

[t09-2] Trevisan, Luca (May 13, 2009), "The Triangle Removal Lemma", in theory

[roth-3] Roth, K. F. (1953), "On certain sets of integers", Journal of the London Mathematical Society, 28 (1): 104–109, doi:10.1112/jlms/s1-28.1.104, MR 0051853

[tao-4] Tao, Terence (2006), "A variant of the hypergraph removal lemma", Journal of Combinatorial Theory, Series A, 113 (7): 1257–1280, arXiv:math/0503572, doi:10.1016/j.jcta.2005.11.006, MR 2259060

[as-5] Alon, Noga; Shapira, Asaf (2004), "Testing subgraphs in directed graphs", Journal of Computer and System Sciences, 69 (3): 353–382, doi:10.1016/j.jcss.2004.04.008, MR 2087940

[rs78-6] Ruzsa, I. Z.; Szemerédi, E. (1978), "Triple systems with no six points carrying three triangles", Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. II, Colloq. Math. Soc. János Bolyai, 18, Amsterdam and New York: North-Holland, pp. 939–945, MR 0519318

[efr-7] Erdős, P.; Frankl, P.; Rödl, V. (1986), "The asymptotic number of graphs not containing a fixed subgraph and a problem for hypergraphs having no exponent", Graphs and Combinatorics, 2 (2): 113–121, doi:10.1007/BF01788085, MR 0932119

[afk-8] Alon, Noga; Fischer, Eldar; Krivelevich, Michael; Szegedy, Mario (2000), "Efficient testing of large graphs", Combinatorica, 20 (4): 451–476, doi:10.1007/s004930070001, MR 1804820

[s03-9] Solymosi, J. (2003), "Note on a generalization of Roth's theorem", Discrete and Computational Geometry, Algorithms and Combinatorics, 25: 825–827, doi:10.1007/978-3-642-55566-4_39, ISBN 978-3-642-62442-1, MR 2038505, S2CID 53973423

[as08-10] Alon, Noga; Shapira, Asaf (2008), "A characterization of the (natural) graph properties testable with one-sided error", SIAM Journal on Computing, 37 (6): 1703–1727, doi:10.1137/06064888X, MR 2386211