Regularity inheritance in pseudorandom graphs

Advancing the sparse regularity method, we prove one‐sided and two‐sided regularity inheritance lemmas for subgraphs of bijumbled graphs, improving on results of Conlon, Fox, and Zhao. These inheritance lemmas also imply improved H‐counting lemmas for subgraphs of bijumbled graphs, for some H.     

An algorithmic hypergraph regularity lemma

Szemerédi 's Regularity Lemma is a powerful tool in graph theory. It asserts that all large graphs admit bounded partitions of their edge sets, most classes of which consist of uniformly distributed edges. The original proof of this result was nonconstructive, and a constructive proof was later given by Alon, Duke, Lefmann, Rödl, and Yuster. Szemerédi's Regularity Lemma was extended to hypergraphs by various authors. Frankl and Rödl gave one such extension in the case of 3‐uniform hypergraphs, which was later extended to k‐uniform hypergraphs by Rödl and Skokan. W.T. Gowers gave another such extension, using a different concept of regularity than that of Frankl, Rödl, and Skokan. Here, we give a constructive proof of a regularity lemma for hypergraphs.     

On smoothed analysis in dense graphs and formulas

We study a model of random graphs, where a random instance is obtained by adding random edges to a large graph of a given density. The research on this model has been started by Bohman and colleagues (Random Struct Algor 3 ; Random Struct Algor 4 ). Here we obtain a sharp threshold for the appearance of a fixed subgraph and for certain Ramsey properties. We also consider a related model of random k‐SAT formulas, where an instance is obtained by adding random k‐clauses to a fixed formula with a given number of clauses, and derive tight bounds for the non‐satisfiability of the thus‐obtained random formula. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

An application of the regularity lemma in generalized Ramsey theory

Given graphs G and H, an edge coloring of G is called an (H,q)‐coloring if the edges of every copy of H ? G together receive at least q colors. Let r(G,H,q) denote the minimum number of colors in a (H,q)‐coloring of G. In 9 Erd?s and Gyárfás studied r(K_n,K_p,q) if p and q are fixed and n tends to infinity. They determined for every fixed p the smallest q (denoted by q_lin) for which r(K_n,K_p,q) is linear in n and the smallest q (denoted by q_quad) for which r(K_n,K_p,q) is quadratic in n. They raised the problem of determining the smallest q for which we have . In this paper by using the Regularity Lemma we show that if , then we have . © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 39–49, 2003     

Extremal results in sparse pseudorandom graphs

Szemerédi's regularity lemma is a fundamental tool in extremal combinatorics. However, the original version is only helpful in studying dense graphs. In the 1990s, Kohayakawa and Rödl proved an analogue of Szemerédi's regularity lemma for sparse graphs as part of a general program toward extending extremal results to sparse graphs. Many of the key applications of Szemerédi's regularity lemma use an associated counting lemma. In order to prove extensions of these results which also apply to sparse graphs, it remained a well-known open problem to prove a counting lemma in sparse graphs.     

Tripartite Ramsey numbers for paths

In this article, we study the tripartite Ramsey numbers of paths. We show that in any two‐coloring of the edges of the complete tripartite graph K(n, n, n) there is a monochromatic path of length (1 ? o(1))2n. Since R(P_2n+1,P_2n+1)=3n, this means that the length of the longest monochromatic path is about the same when two‐colorings of K_3n and K(n, n, n) are considered. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 164–174, 2007     

Strong edge colorings of uniform graphs

For a graph G=(V(G),E(G)), a strong edge coloring of G is an edge coloring in which every color class is an induced matching. The strong chromatic index of G, χ_s(G), is the smallest number of colors in a strong edge coloring of G. The strong chromatic index of the random graph G(n,p) was considered in Discrete Math. 281 (2004) 129, Austral. J. Combin. 10 (1994) 97, Austral. J. Combin. 18 (1998) 219 and Combin. Probab. Comput. 11 (1) (2002) 103. In this paper, we consider χ_s(G) for a related class of graphs G known as uniform or ε-regular graphs. In particular, we prove that for 0<ε?d<1, all (d,ε)-regular bipartite graphs G=(U∪V,E) with |U|=|V|?n₀(d,ε) satisfy χ_s(G)?ζ(ε)Δ(G)², where ζ(ε)→0 as ε→0 (this order of magnitude is easily seen to be best possible). Our main tool in proving this statement is a powerful packing result of Pippenger and Spencer (Combin. Theory Ser. A 51(1) (1989) 24).     

Coloring uniform hypergraphs with few colors

Let m(r, k) denote the minimum number of edges in an r‐uniform hypergraph that is not k‐colorable. We give a new lower bound on m(r, k) for fixed k and large r. Namely, we prove that if k ≥ 2ⁿ, then m(r, k) ≥ ?(k)k^r(r/ln r)^n/(n+1). © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 2004     

A variant of the hypergraph removal lemma

Recent work of Gowers [T. Gowers, A new proof of Szemerédi's theorem, Geom. Funct. Anal. 11 (2001) 465-588] and Nagle, Rödl, Schacht, and Skokan [B. Nagle, V. Rödl, M. Schacht, The counting lemma for regular k-uniform hypergraphs, Random Structures Algorithms, in press; V. Rödl, J. Skokan, Regularity lemma for k-uniform hypergraphs, Random Structures Algorithms, in press; V. Rödl, J. Skokan, Applications of the regularity lemma for uniform hypergraphs, preprint] has established a hypergraph removal lemma, which in turn implies some results of Szemerédi [E. Szemerédi, On sets of integers containing no k elements in arithmetic progression, Acta Arith. 27 (1975) 299-345], and Furstenberg and Katznelson [H. Furstenberg, Y. Katznelson, An ergodic Szemerédi theorem for commuting transformations, J. Anal. Math. 34 (1978) 275-291] concerning one-dimensional and multidimensional arithmetic progressions, respectively. In this paper we shall give a self-contained proof of this hypergraph removal lemma. In fact we prove a slight strengthening of the result, which we will use in a subsequent paper [T. Tao, The Gaussian primes contain arbitrarily shaped constellations, preprint] to establish (among other things) infinitely many constellations of a prescribed shape in the Gaussian primes.     

On the editing distance of graphs

An edge‐operation on a graph G is defined to be either the deletion of an existing edge or the addition of a nonexisting edge. Given a family of graphs , the editing distance from G to is the smallest number of edge‐operations needed to modify G into a graph from . In this article, we fix a graph H and consider Forb(n, H), the set of all graphs on n vertices that have no induced copy of H. We provide bounds for the maximum over all n‐vertex graphs G of the editing distance from G to Forb(n, H), using an invariant we call the binary chromatic number of the graph H. We give asymptotically tight bounds for that distance when H is self‐complementary and exact results for several small graphs H. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:123–138, 2008     

Almost Given Length Cycles in Digraphs

A digraph is called k-cyclic if it cannot be made acyclic by removing less than k arcs. It is proved that for every ε > 0 there are constants K and δ so that for every d ∈ (0, δn), every ε n²-cyclic digraph with n vertices contains a directed cycle whose length is between d and d + K. A more general result of the same form is obtained for blow-ups of directed cycles.     

What is the furthest graph from a hereditary property?

For a graph property P, the edit distance of a graph G from P, denoted E_P(G), is the minimum number of edge modifications (additions or deletions) one needs to apply to G to turn it into a graph satisfying P. What is the furthest graph on n vertices from P and what is the largest possible edit distance from P? Denote this maximal distance by ed(n,P). This question is motivated by algorithmic edge‐modification problems, in which one wishes to find or approximate the value of E_P(G) given an input graph G. A monotone graph property is closed under removal of edges and vertices. Trivially, for any monotone property, the largest edit distance is attained by a complete graph. We show that this is a simple instance of a much broader phenomenon. A hereditary graph property is closed under removal of vertices. We prove that for any hereditary graph property P, a random graph with an edge density that depends on P essentially achieves the maximal distance from P, that is: ed(n,P) = E_P(G(n,p(P))) + o(n²) with high probability. The proofs combine several tools, including strengthened versions of the Szemerédi regularity lemma, properties of random graphs and probabilistic arguments. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

The averaging lemma

Averaging lemmas deduce smoothness of velocity averages, such as

from properties of . A canonical example is that is in the Sobolev space whenever and are in . The present paper shows how techniques from Harmonic Analysis such as maximal functions, wavelet decompositions, and interpolation can be used to prove versions of the averaging lemma. For example, it is shown that implies that is in the Besov space , . Examples are constructed using wavelet decompositions to show that these averaging lemmas are sharp. A deeper analysis of the averaging lemma is made near the endpoint .

     

Loose Hamilton cycles in hypergraphs

We prove that any k-uniform hypergraph on n vertices with minimum degree at least contains a loose Hamilton cycle. The proof strategy is similar to that used by Kühn and Osthus for the 3-uniform case. Though some additional difficulties arise in the k-uniform case, our argument here is considerably simplified by applying the recent hypergraph blow-up lemma of Keevash.     

Separation properties of induced I(L)-topological spaces

The concept of induced I(L)-topological spaces has been introduced by Kubiak (Ph.D. Thesis, UAM, Poznan, 1985) and independently by Wang (Kexue Tongbao 34 (5) (1989) 333). In this paper, the separation properties in the sense of Hutton–Reilly of induced I(L)-topological spaces are investigated. The main result of the paper is a characterization of L-topological spaces by means of the appropriate Hutton–Reilly separation properties of its induced I(L)-topological space.     

On an anti‐Ramsey problem of Burr,Erdős,Graham, and T. Sós

Given a graph L, in this article we investigate the anti‐Ramsey number χ_S(n,e,L), defined to be the minimum number of colors needed to edge‐color some graph G(n,e) with n vertices and e edges so that in every copy of L in G all edges have different colors. We call such a copy of L totally multicolored (TMC). In 7 among many other interesting results and problems, Burr, Erd?s, Graham, and T. Sós asked the following question: Let L be a connected bipartite graph which is not a star. Is it true then that In this article, we prove a slightly weaker statement, namely we show that the statement is true if L is a connected bipartite graph, which is not a complete bipartite graph. © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 147–156, 2006     

A multipartite Ramsey number for odd cycles

In this article we study multipartite Ramsey numbers for odd cycles. Our main result is the proof that a conjecture of Gyárfás et al. (J Graph Theory 61 (2009), 12–21), holds for graphs with a large enough number of vertices. Precisely, there exists n₀ such that if n?n₀ is a positive odd integer then any two‐coloring of the edges of the complete five‐partite graph K_{(n ? 1)/2, (n ? 1)/2, (n ? 1)/2, (n ? 1)/2, 1} contains a monochromatic cycle of length n. © 2011 Wiley Periodicals, Inc. J Graph Theory