Yoshiharu Kohayakawa

Szemerédi's regularity lemma for sparse graphs

A remarkable lemma of Szemeredi asserts that, very roughly speaking, any dense graph can be decomposed into a bounded number of pseudorandom bipartite graphs. This far-reaching result has proved to play a central role in many areas of combinatorics, both ‘pure’ and ‘algorithmic.’ The quest for an equally powerful variant of this lemma for sparse graphs has not yet been successful, but some progress has been achieved recently. The aim of this note is to report on the successes so far.

On K 4 -Free Subgraphs of Random Graphs.

For 0<γ≤1 and graphsG andH, writeG→γH if any γ-proportion of the edges ofG spans at least one copy ofH inG. As customary, writeKr for the complete graph onr vertices. We show that for every fixed real η>0 there exists a constantC=C(η) such that almost every random graphGn,p withp=p(n)≥Cn−2/5 satisfiesGn,p→2/3+ηK4. The proof makes use of a variant of Szemerédis regularity lemma for sparse graphs and is based on a certain superexponential estimate for the number of pseudo-random tripartite graphs whose triangles are not too well distributed. Related results and a general conjecture concerningH-free subgraphs of random graphs in the spirit of the Erdős-Stone theorem are discussed.

Hypergraphs, Quasi-randomness, and Conditions for Regularity

Haviland and Thomason and Chung and Graham were the first to investigate systematically some properties of quasi-random hypergraphs. In particular, in a series of articles, Chung and Graham considered several quite disparate properties of random-like hypergraphs of density 1/2 and proved that they are in fact equivalent. The central concept in their work turned out to be the so called deviation of a hypergraph. They proved that having small deviation is equivalent to a variety of other properties that describe quasi-randomness. In this paper, we consider the concept of discrepancy for k-uniform hypergraphs with an arbitrary constant density d (0

Finding Skew Partitions Efficiently

A skew partition as defined by Chvatal is a partition of the vertex set of a graph into four nonempty parts A,B,C,D such that there are all possible edges between A and B and no edges between C and D. We present a polynomial-time algorithm for testing whether a graph admits a skew partition. Our algorithm solves the more general list skew partition problem, where the input contains, for each vertex, a list containing some of the labels A,B,C,D of the four parts. Our polynomial-time algorithm settles the complexity of the original partition problem proposed by Chvatal in 1985 and answers a recent question of Feder, Hell, Klein, and Motwani.

Regular pairs in sparse random graphs I

We consider bipartite subgraphs of sparse random graphs that are regular in the sense of Szemeredi and, among other things, show that they must satisfy a certain local pseudorandom property. This property and its consequences turn out to be useful when considering embedding problems in subgraphs of sparse random graphs.

The Induced Size-Ramsey Number of Cycles

For a graph H and an integer r ≥ 2, the induced r-size-Ramsey number of H is defined to be the smallest integer m for which there exists a graph G with m edges with the following property: however one colours the edges of G with r colours, there always exists a monochromatic induced subgraph H ′ of G that is isomorphic to H . This is a concept closely related to the classical r -size-Ramsey number of Erdős, Faudree, Rousseau and Schelp, and to the r -induced Ramsey number, a natural notion that appears in problems and conjectures due to, among others, Graham and Rodl, and Trotter. Here, we prove a result that implies that the induced r -size-Ramsey number of the cycle C l is at most c r l for some constant c r that depends only upon r . Thus we settle a conjecture of Graham and Rodl, which states that the above holds for the path P l of order l and also generalise in part a result of Bollobas, Burr and Reimer that implies that the r -size Ramsey number of the cycle C l is linear in l Our method of proof is heavily based on techniques from the theory of random graphs and on a variant of the powerful regularity lemma of Szemeredi.

An Extremal Problem For Random Graphs And The Number Of Graphs With Large Even-Girth

C2k-free subgraph of a random graph may have, obtaining best possible results for a range of p=p(n). Our estimates strengthen previous bounds of Füredi [12] and Haxell, Kohayakawa, and Łuczak [13]. Two main tools are used here: the first one is an upper bound for the number of graphs with large even-girth, i.e., graphs without short even cycles, with a given number of vertices and edges, and satisfying a certain additional pseudorandom condition; the second tool is the powerful result of Ajtai, Komlós, Pintz, Spencer, and Szemerédi [1] on uncrowded hypergraphs as given by Duke, Lefmann, and Rödl [7].

An Optimal Algorithm for Checking Regularity

We present a deterministic algorithm

Weak hypergraph regularity and linear hypergraphs

{\cal A}

Szemerédi’s Regularity Lemma and Quasi-randomness

that, in O(m2) time, verifies whether a given m by m bipartite graph G is regular, in the sense of Szemeredi [Regular partitions of graphs, in Problemes Combinatoires et Theorie des Graphes (Orsay, 1976), Colloques Internationaux CNRS 260, CNRS, Paris, 1978, pp. 399--401]. In the case in which G is not regular enough, our algorithm outputs a witness to this irregularity. Algorithm