Dmitry A. Shabanov

Improved algorithms for colorings of simple hypergraphs and applications

The paper deals with extremal problems concerning colorings of hypergraphs. By using a random recoloring algorithm we show that any n-uniform simple (i.e. every two distinct edges share at most one vertex) hypergraph H with maximum edge degree at most Δ ( H ) ≤ c ? n r n - 1 is r-colorable, where c 0 is an absolute constant.As an application of our proof technique we establish a new lower bound for Van der Waerden number W ( n , r ) , the minimum N such that in any r-coloring of the set { 1 , ? , N } there exists a monochromatic arithmetic progression of length n. We show that W ( n , r ) c ? r n - 1 , for some absolute constant c 0 .

The Distribution of Second Degrees in the Buckley–Osthus Random Graph Model

In this article we consider a well-known generalization of the Barabási and Albert preferential attachment model—the Buckley–Osthus model. Buckley and Osthus proved that in this model, the degree sequence has a power law distribution. As a natural (and arguably more interesting) next step, we study the second degrees of vertices. Roughly speaking, the second degree of a vertex is the number of vertices at distance two from the given vertex. The distribution of second degrees is of interest because it is a good approximation of PageRank, where the importance of a vertex is measured by taking into account the popularity of its neighbors. We prove that the second degrees also obey a power law. More precisely, we estimate the expectation of the number of vertices with the second degree greater than or equal to k and prove the concentration of this random variable around its expectation using the now-famous Talagrands concentration inequality over product spaces. As far as we know, this is the only application of Talagrands inequality to random web graphs where the (preferential attachment) edges are not defined over a product distribution, making the application nontrivial and requiring a certain degree of novelty.

Equitable two-colorings of uniform hypergraphs

An equitable two-coloring of a hypergraph H = ( V , E ) is a proper vertex two-coloring such that the cardinalities of color classes differ by at most one. In connection with the property B problem Radhakrishnan and Srinivasan proved that if H is a k -uniform hypergraph with maximum vertex degree Δ ( H ) satisfying Δ ( H ) ≤ c 2 k - 1 k ln k for some absolute constant c 0 , then H is 2-colorable. By using the Lovasz Local Lemma for negatively correlated events and the random recoloring method we prove that if H either is a simple hypergraph or has a lot of vertices, then under the same condition on the maximum vertex degree it has an equitable coloring with two colors. We also obtain a general result for equitable colorings of partial Steiner systems.

On the Problem of Erdős and Hajnal in the Case of List Colorings

Abstract We deal with the classical problem of Erdős and Hajnal in hypergraph theory and its generalization concerning the list colorings of hypergraphs. Let m ( n , k ) ( m list ( n , k ) ) denote the minimum number of edges in an n-uniform hypergraph with chromatic (list chromatic) number k + 1 . We obtained some new lower bounds for m ( n , k ) and m list ( n , k ) which improved previous results for some values of parameters n and k.

Colorings of hypergraphs with large number of colors

The paper deals with the well-known problem of Erdźs and Hajnal concerning colorings of uniform hypergraphs and some related questions. Let m ( n , r ) denote the minimum possible number of edges in an n -uniform non- r -colorable hypergraph. We show that for r n , c 1 n ln n ≤ m ( n , r ) r n ≤ C 1 n 3 ln n , where c 1 , C 1 0 are some absolute constants. Moreover, we obtain similar bounds for d ( n , r ) , which is equal to the minimum possible value of the maximum edge degree in an n -uniform non- r -colorable hypergraph. If r n , then c 2 n ln n ≤ d ( n , r ) r n - 1 ≤ C 2 n 3 ln n , where c 2 , C 2 0 are some other absolute constants.

Around Erdős-Lovász problem on colorings of non-uniform hypergraphs

The work deals with combinatorial problems concerning colorings of non-uniform hypergraphs. Let H = ( V , E ) be a hypergraph with minimum edge-cardinality n . We show that if H is a simple hypergraph (i.e.?every two distinct edges have at most one common vertex) and ? e ? E r 1 - | e | ≤ c n , for some absolute constant c 0 , then H is r -colorable. We also obtain a stronger result for triangle-free simple hypergraphs by proving that if H is a simple triangle-free hypergraph and ? e ? E r 1 - | e | ≤ c ? n , for some absolute constant c 0 , then H is r -colorable.

Colorings of Uniform Hypergraphs with Large Girth

This work deals with a wellknown extremal com� binatorial problem concerning colorings of uniform hypergraphs with large girth. First, we recall the basic concepts from hypergraph theory. A hypergraph is a pair of sets H =( V, E), where V is a finite set known as the vertex set of the hypergraph and E is a collection of subsets of V, with these subsets called the edges of the hypergraph. A hypergraph is nuniform if each of its edges contains exactly n vertices. A coloring of the vertex set V in H = (V, E) is said to be proper if all the edges in E are not monochro� matic. The chromatic number of H is the minimum number of colors required for a proper coloring of its vertex set. The chromatic number of H is denoted by χ(H). A cycle of length k in H = (V, E) is an alternating sequence v 0 , e 1 , v 1 , …, e k , v k = v 0 consisting of k dif� ferent vertices v0, v1 ,… , vk -1 and k different edges e1, e2, …, ek such that vi -1 ∈ ei and vi ∈ ei for any i = 1, 2, …, k. The length of the minimum cycle in H is called its girth and is denoted by g(H). The degree of a vertex v in H is the number of edges in H that contain v. The maximum vertex degree in a hypergraph is denoted by Δ(H). The study of the interrelation between the chro� matic number, girth, and maximum vertex degree in nuniform hypergraphs was begun in Erdos and Lovaszs classical work (1). They showed that, if an n� uniform hypergraph has a large chromatic number, then its maximum vertex degree cannot be very low. Thereby, they motivated the study of the extremal value Δ(n, r, s) equal to the minimum possible value of the maximum vertex degree of am nuniform hyper� graph with a chromatic number greater than r and girth greater than s. Formally, Δ(n, r, s) is defined as This definition implies that

General Independence Sets in Random Strongly Sparse Hypergraphs

AbstractWe analyze the asymptotic behavior of the j-independence number of a random k-uniform hypergraph H(n, k, p) in the binomial model. We prove that in the strongly sparse case, i.e., where

Panchromatic 3-coloring of a random hypergraph

p = c/\left( \begin{gathered} n - 1 \hfill \\ k - 1 \hfill \\ \end{gathered} \right)