scholarly journals NOTES ON THE STABLE REGULARITY LEMMA

2021 ◽  
pp. 1-7
Author(s):  
M. MALLIARIS ◽  
S. SHELAH
Keyword(s):  
Regularity Lemma

scholarly journals Hereditary quasirandomness without regularity

2017 ◽  
Vol 164 (3) ◽  
pp. 385-399 ◽  
Author(s):  
DAVID CONLON ◽  
JACOB FOX ◽  
BENNY SUDAKOV
Keyword(s):  
Alternative Proof ◽  
Regularity Lemma ◽  
Original Proof ◽  
Vertex Graph

AbstractA result of Simonovits and Sós states that for any fixed graph H and any ε > 0 there exists δ > 0 such that if G is an n-vertex graph with the property that every S ⊆ V(G) contains pe(H) |S|v(H) ± δ nv(H) labelled copies of H, then G is quasirandom in the sense that every S ⊆ V(G) contains $\frac{1}{2}$p|S|2± ε n2 edges. The original proof of this result makes heavy use of the regularity lemma, resulting in a bound on δ−1 which is a tower of twos of height polynomial in ε−1. We give an alternative proof of this theorem which avoids the regularity lemma and shows that δ may be taken to be linear in ε when H is a clique and polynomial in ε for general H. This answers a problem raised by Simonovits and Sós.

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 Hypergraph Independent Sets

2012 ◽  
Vol 22 (1) ◽  
pp. 9-20 ◽  
Author(s):  
JONATHAN CUTLER ◽  
A. J. RADCLIFFE
Keyword(s):  
Upper Bound ◽  
Independent Set ◽  
Independent Sets ◽  
Extremal Problems ◽  
Regularity Lemma ◽  
Uniform Hypergraph ◽  
Fixed Size

The study of extremal problems related to independent sets in hypergraphs is a problem that has generated much interest. There are a variety of types of independent sets in hypergraphs depending on the number of vertices from an independent set allowed in an edge. We say that a subset of vertices isj-independentif its intersection with any edge has size strictly less thanj. The Kruskal–Katona theorem implies that in anr-uniform hypergraph with a fixed size and order, the hypergraph with the mostr-independent sets is the lexicographic hypergraph. In this paper, we use a hypergraph regularity lemma, along with a technique developed by Loh, Pikhurko and Sudakov, to give an asymptotically best possible upper bound on the number ofj-independent sets in anr-uniform hypergraph.

Covering Two-Edge-Coloured Complete Graphs with Two Disjoint Monochromatic Cycles

2008 ◽  
Vol 17 (4) ◽  
pp. 471-486 ◽  
Author(s):  
PETER ALLEN
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Complete Graphs ◽  
Regularity Lemma ◽  
Alternative Method ◽  
Vertex Disjoint ◽  
Edge Colouring

In 1998 Łuczak Rödl and Szemerédi [7] proved, by means of the Regularity Lemma, that there exists n0 such that, for any n ≥ n0 and two-edge-colouring of Kn, there exists a pair of vertex-disjoint monochromatic cycles of opposite colours covering the vertices of Kn. In this paper we make use of an alternative method of finding useful structure in a graph, leading to a proof of the same result with a much smaller value of n0. The proof gives a polynomial-time algorithm for finding the two cycles.

scholarly journals Limits of kernel operators and the spectral regularity lemma

2011 ◽  
Vol 32 (7) ◽  
pp. 1156-1167 ◽  
Author(s):  
Balázs Szegedy
Keyword(s):  
Regularity Lemma ◽  
Kernel Operators

scholarly journals A Characterization for Sparse $\varepsilon$-Regular Pairs

10.37236/923 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Stefanie Gerke ◽  
Angelika Steger
Keyword(s):  
Bipartite Graph ◽  
Regular Graph ◽  
Bipartite Graphs ◽  
Regularity Lemma ◽  
Sparse Graphs

We are interested in $(\varepsilon)$-regular bipartite graphs which are the central objects in the regularity lemma of Szemerédi for sparse graphs. A bipartite graph $G=(A\uplus B,E)$ with density $p={|E|}/({|A||B|})$ is $(\varepsilon)$-regular if for all sets $A'\subseteq A$ and $B'\subseteq B$ of size $|A'|\geq \varepsilon|A|$ and $|B'|\geq \varepsilon |B|$, it holds that $\left| {e_G(A',B')}/{(|A'||B'|)}- p\right| \leq \varepsilon p$. In this paper we prove a characterization for $(\varepsilon)$-regularity. That is, we give a set of properties that hold for each $(\varepsilon)$-regular graph, and conversely if the properties of this set hold for a bipartite graph, then the graph is $f(\varepsilon)$-regular for some appropriate function $f$ with $f(\varepsilon)\rightarrow 0$ as $\varepsilon\rightarrow 0$. The properties of this set concern degrees of vertices and common degrees of vertices with sets of size $\Theta(1/p)$ where $p$ is the density of the graph in question.

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