scholarly journals A Simple Algorithm for Constructing Szemerédi's Regularity Partition

10.37236/1449 ◽  
1999 ◽  
Vol 6 (1) ◽  
Author(s):  
Alan Frieze ◽  
Ravi Kannan
Keyword(s):  
Simple Algorithm ◽  
Singular Values ◽  
Regularity Lemma ◽  
Constructive Version ◽  
Szemeredi's Regularity Lemma

We give a simple constructive version of Szemerédi's Regularity Lemma, based on the computation of singular values of matrices.

scholarly journals Tilings in Randomly Perturbed Dense Graphs

2018 ◽  
Vol 28 (2) ◽  
pp. 159-176 ◽  
Author(s):  
JÓZSEF BALOGH ◽  
ANDREW TREGLOWN ◽  
ADAM ZSOLT WAGNER
Keyword(s):  
High Probability ◽  
Minimum Degree ◽  
Graph Model ◽  
Regularity Lemma ◽  
Arbitrary Graph ◽  
Random Graph Model ◽  
Dense Graphs ◽  
Szemeredi's Regularity Lemma ◽  
Vertex Disjoint ◽  
Special Case

A perfect H-tiling in a graph G is a collection of vertex-disjoint copies of a graph H in G that together cover all the vertices in G. In this paper we investigate perfect H-tilings in a random graph model introduced by Bohman, Frieze and Martin [6] in which one starts with a dense graph and then adds m random edges to it. Specifically, for any fixed graph H, we determine the number of random edges required to add to an arbitrary graph of linear minimum degree in order to ensure the resulting graph contains a perfect H-tiling with high probability. Our proof utilizes Szemerédi's Regularity Lemma [29] as well as a special case of a result of Komlós [18] concerning almost perfect H-tilings in dense graphs.

scholarly journals Stability for Vertex Cycle Covers

10.37236/5185 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
József Balogh ◽  
Frank Mousset ◽  
Jozef Skokan
Keyword(s):  
Minimum Degree ◽  
Independent Set ◽  
Stability Result ◽  
Natural Generalization ◽  
Regularity Lemma ◽  
Property A ◽  
Vertex Graph ◽  
Szemeredi's Regularity Lemma ◽  
Cycle Covers ◽  
Positive Integers

In 1996 Kouider and Lonc proved the following natural generalization of Dirac's Theorem: for any integer $k\geq 2$, if $G$ is an $n$-vertex graph with minimum degree at least $n/k$, then there are $k-1$ cycles in $G$ that together cover all the vertices.This is tight in the sense that there are $n$-vertex graphs that have minimum degree $n/k-1$ and that do not contain $k-1$ cycles with this property. A concrete example is given by $I_{n,k} = K_n\setminus K_{(k-1)n/k+1}$ (an edge-maximal graph on $n$ vertices with an independent set of size $(k-1)n/k+1$). This graph has minimum degree $n/k-1$ and cannot be covered with fewer than $k$ cycles. More generally, given positive integers $k_1,\dotsc,k_r$ summing to $k$, the disjoint union $I_{k_1n/k,k_1}+ \dotsb + I_{k_rn/k,k_r}$ is an $n$-vertex graph with the same properties.In this paper, we show that there are no extremal examples that differ substantially from the ones given by this construction. More precisely, we obtain the following stability result: if a graph $G$ has $n$ vertices and minimum degree nearly $n/k$, then it either contains $k-1$ cycles covering all vertices, or else it must be close (in ‘edit distance') to a subgraph of $I_{k_1n/k,k_1}+ \dotsb + I_{k_rn/k,k_r}$, for some sequence $k_1,\dotsc,k_r$ of positive integers that sum to $k$.Our proof uses Szemerédi's Regularity Lemma and the related machinery.

scholarly journals Szemerédi's Regularity Lemma via Martingales

10.37236/5585 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Pandelis Dodos ◽  
Vassilis Kanellopoulos ◽  
Thodoris Karageorgos
Keyword(s):  
Random Variables ◽  
Regularity Lemma ◽  
Martingale Difference ◽  
Difference Sequences ◽  
Szemeredi's Regularity Lemma ◽  
Probabilistic Version

We prove a variant of the abstract probabilistic version of Szemerédi's regularity lemma, due to Tao, which applies to a number of structures (including graphs, hypergraphs, hypercubes, graphons, and many more) and works for random variables in $L_p$ for any $p>1$. Our approach is based on martingale difference sequences.

scholarly journals On the Size-Ramsey Number of Cycles

2019 ◽  
Vol 28 (06) ◽  
pp. 871-880
Author(s):  
R. Javadi ◽  
F. Khoeini ◽  
G. R. Omidi ◽  
A. Pokrovskiy
Keyword(s):  
Upper Bounds ◽  
Ramsey Number ◽  
Alternative Proof ◽  
Regularity Lemma ◽  
Ramsey Numbers ◽  
Szemeredi's Regularity Lemma ◽  
Number Of Cycles ◽  
Image Position ◽  
Edge Colouring ◽  
Monochromatic Copy

AbstractFor given graphs G1,…, Gk, the size-Ramsey number $\hat R({G_1}, \ldots ,{G_k})$ is the smallest integer m for which there exists a graph H on m edges such that in every k-edge colouring of H with colours 1,…,k, H contains a monochromatic copy of Gi of colour i for some 1 ≤ i ≤ k. We denote $\hat R({G_1}, \ldots ,{G_k})$ by ${\hat R_k}(G)$ when G1 = ⋯ = Gk = G.Haxell, Kohayakawa and Łuczak showed that the size-Ramsey number of a cycle Cn is linear in n, ${\hat R_k}({C_n}) \le {c_k}n$ for some constant ck. Their proof, however, is based on Szemerédi’s regularity lemma so no specific constant ck is known.In this paper, we give various upper bounds for the size-Ramsey numbers of cycles. We provide an alternative proof of ${\hat R_k}({C_n}) \le {c_k}n$ , avoiding use of the regularity lemma, where ck is exponential and doubly exponential in k, when n is even and odd, respectively. In particular, we show that for sufficiently large n we have ${\hat R_2}({C_n}) \le {10^5} \times cn$ , where c = 6.5 if n is even and c = 1989 otherwise.

The Blow-up Lemma

1999 ◽  
Vol 8 (1-2) ◽  
pp. 161-176 ◽  
Author(s):  
JÁNOS KOMLÓS
Keyword(s):  
Graph Theory ◽  
Blow Up ◽  
Extremal Graph ◽  
Extremal Graph Theory ◽  
Regularity Lemma ◽  
Sparse Graphs ◽  
Dense Graphs ◽  
Recent Developments ◽  
Szemeredi's Regularity Lemma

Extremal graph theory has a great number of conjectures concerning the embedding of large sparse graphs into dense graphs. Szemerédi's Regularity Lemma is a valuable tool in finding embeddings of small graphs. The Blow-up Lemma, proved recently by Komlós, Sárközy and Szemerédi, can be applied to obtain approximate versions of many of the embedding conjectures. In this paper we review recent developments in the area.

Regular Partitions of Hypergraphs: Regularity Lemmas

2007 ◽  
Vol 16 (6) ◽  
pp. 833-885 ◽  
Author(s):  
VOJTĚCH RÖDL ◽  
MATHIAS SCHACHT
Keyword(s):  
Regularity Lemma ◽  
Szemeredi's Regularity Lemma ◽  
User Friendly ◽  
Regular Partitions

Szemerédi's regularity lemma for graphs has proved to be a powerful tool with many subsequent applications. The objective of this paper is to extend the techniques developed by Nagle, Skokan, and the authors and obtain a stronger and more ‘user-friendly’ regularity lemma for hypergraphs.

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