Vojtech Rödl

The algorithmic aspects of the regularity lemma

Abstract The regularity lemma of Szemeredi asserts that every graph can be partitioned in a certain regular way. This result has numerous applications, but its known proof is not algorithmic. Here we first demonstrate the computational difficulty of finding a regular partition; we show that deciding if a given partition of an input graph satisfies the properties guaranteed by the lemma is co-NP-complete. However, we also prove that despite this difficulty the lemma can be made constructive; we show how to obtain, for any input graph, a partition with the properties guaranteed by the lemma, efficiently. The desired partition, for an n -vertex graph, can be found in time O ( M ( n )), where M ( n ) = O ( n 2.376 ) is the time needed to multiply two n by n matrices with 0, 1-entries over the integers. The algorithm can be parallelized and implemented in NC 1 . Besides the curious phenomenon of exhibiting a natural problem in which the search for a solution is easy whereas the decision if a given instance is a solution is difficult (if P and NP differ), our constructive version of the regularity lemma supplies efficient sequential and parallel algorithms for many problems, some of which are naturally motivated by the study of various graph embedding and graph coloring problems.

The Ramsey number of a graph with bounded maximum degree

Abstract The Ramsey number of a graph G is the least number t for which it is true that whenever the edges of the complete graph on t vertices are colored in an arbitrary fashion using two colors, say red and blue, then it is always the case that either the red subgraph contains G or the blue subgraph contains G . A conjecture of P. Erdos and S. Burr is settled in the affirmative by proving that for each d ≥ 1, there exists a constant c so that if G is any graph on n vertices with maximum degree d , then the Ramsey number of G is at most cn .

Geometrical realization of set systems and probabilistic communication complexity

Let d = d(n) be the minimum d such that for every sequence of n subsets F1, F2, . . . , Fn of {1, 2, . . . , n} there exist n points P1, P2, . . . , Pn and n hyperplanes H1, H2 .... , Hn in Rd such that Pj lies in the positive side of Hi iff j ∈ Fi. Then n/32 ≤ d(n) ≤ (1/2 + 0(1)) ¿ n. This implies that the probabilistic unbounded-error 2-way complexity of almost all the Boolean functions of 2p variables is between p-5 and p, thus solving a problem of Yao and another problem of Paturi and Simon. The proof of (1) combines some known geometric facts with certain probabilistic arguments and a theorem of Milnor from real algebraic geometry.

Large triangle-free subgraphs in graphs withoutK4

It is shown that for arbitrary positiveε there exists a graph withoutK4 and so that all its subgraphs containing more than 1/2 +ε portion of its edges contain a triangle (Theorem 2). This solves a problem of Erdös and Nešetřil. On the other hand it is proved that such graphs have necessarily low edge density (Theorem 4).Theorem 3 which is needed for the proof of Theorem 2 is an analog of Goodmans theorem [8], it shows that random graphs behave in some respect as sparse complete graphs.Theorem 5 shows the existence of a graph on less than 1012 vertices, withoutK4 and which is edge-Ramsey for triangles.

Regular Partitions of Hypergraphs: Regularity Lemmas

Szemeredis 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.

Dirac-Type Questions For Hypergraphs — A Survey (Or More Problems For Endre To Solve)

Dedicated to Endre Szemeredi on the occasion of his 70th birthday In 1952 Dirac [8] proved a celebrated theorem stating that if the minimum degree δ(G) in an n-vertex graph G is at least n/2 then G contains a Hamiltonian cycle. In 1999, Katona and Kierstead initiated a new stream of research devoted to studying similar questions for hypergraphs, and subsequently, for perfect matchings. A pivotal role in achieving some of the most important results in both these areas was played by Endre Szemeredi. In this survey we present the current state-of-art and pose some open problems.

The Uniformity Lemma for hypergraphs

In 1973, E. Szemeredi proved a theorem which found numerous applications in extremal combinatorial problems—The Uniformity Lemma for Graphs. Here we consider an extension of Szemeredis theorem tor-uniform hypergraphs.

On graphs with small subgraphs of large chromatic number

Bollobás, Erdös, Simonovits, and Szemerédi conjectured [1] that for each positive constantc there exists a constantg(c) such that ifG is any graph which cannot be made 3-chromatic by the omission ofcn2 edges, thenG contains a 4-chromatic subgraph with at mostg(c) vertices. Here we establish the following generalization which was suggested by Erdös [2]: For each positive constantc and positive integerk there exist positive integersfk(c) andno such that ifG is any graph with more thanno vertices having the property that the chromatic number ofG cannot be made less thank by the omission of at mostcn2 edges, thenG contains ak-chromatic subgraph with at mostfk(c) vertices.

Large matchings in uniform hypergraphs and the conjectures of Erdős and Samuels

In this paper we study degree conditions which guarantee the existence of perfect matchings and perfect fractional matchings in uniform hypergraphs. We reduce this problem to an old conjecture by Erdos on estimating the maximum number of edges in a hypergraph when the (fractional) matching number is given, which we are able to solve in some special cases using probabilistic techniques. Based on these results, we obtain some general theorems on the minimum d-degree ensuring the existence of perfect (fractional) matchings. In particular, we asymptotically determine the minimum vertex degree which guarantees a perfect matching in 4-uniform and 5-uniform hypergraphs. We also discuss an application to a problem of finding an optimal data allocation in a distributed storage system.

On graphs with linear Ramsey numbers

In this paper, we prove that any graph G with maximum degree