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Papers by Sergei Bezrukov

We present here a description of all solutions of the isoperimetric problem in Hamming space of s... more We present here a description of all solutions of the isoperimetric problem in Hamming space of some special cardinalities. The number of these cardinalities equals 2 n?1 . Let B n denotes the vertex set of the n?dimensional unit cube with Hamming metric and A B n . Denote by S n k ( ) the sphere of radius k centered in 2 B n . We call a point 2 A the inner point of a set A if S n 1 ( ) A and the boundary point of A in the opposite case. Denote by P(A)?(A) the collection of all inner (boundary) points of A.

We present a specification of all maximum subsets of the n-cube B n with respect to a given diame... more We present a specification of all maximum subsets of the n-cube B n with respect to a given diameter and show relations between this problem and the discrete isoperimetric problem. As a corollary, for any number m, 1 ≤ m ≤ 2 n , we specify an m-element subset of B n with minimal possible diameter. We also present a simple proof for a theorem of Katona on intersecting families . * This paper is a translation from Russian of the author's article published in Problems of Information Transmission, v. XXIII (1987), No 1, 106-109.

Combinatorics, Probability and Computing, 1994

ABSTRACT We consider the oriented binary tree and the oriented hypercube. The tree edges are orie... more ABSTRACT We consider the oriented binary tree and the oriented hypercube. The tree edges are oriented from the root to the leaves, while the orientation of the cube edges is induced by the direction from 0 to 1 in the coordinatewise form. The problem is to embed such a tree with l levels into the oriented n-cube as an oriented subgraph, for minimal possible n. A new approach to such problems is presented, which improves the known upper bound n/l ≤ 3/2 given by Havel [1] to n/l ≤ 4/3 + o(1) as l → ∞.

Lecture Notes in Computer Science, 1999

ABSTRACT . In this paper we introduce a new order on the set of n- dimensional tuples and prove t... more ABSTRACT . In this paper we introduce a new order on the set of n- dimensional tuples and prove that this order preserves nestedness in the edge isoperimetric problem for the graph P n , dened as the n th cartesian power of the well-known Petersen graph. Thus, we show, that there is a graph for which powers the solution of the edge isoperimetric problem preserve nestedness and it is dierent from the lexicographic order. With respect to this result we determine the cutwidth and wirelength of P n . These results are then generalized to the cartesian product of P n and the m-dimensional binary hypercube. 1 Introduction The subject of the paper is the edge-isoperimetric problem, which consists of nding a subset of vertices of a given graph, such that the number of edges separating this subset from its complement, also called edge cut, has minimal size among all subsets of the same cardinality. For a graph G = (VG ; EG ) with vertex set VG , edge set EG and A VG denote G (A) = f(u...

Macaulay posets are posets for which there is an analogue of the classical Kruskal- Katona theore... more Macaulay posets are posets for which there is an analogue of the classical Kruskal- Katona theorem for nite sets. These posets are of great importance in many branches of combinatorics and have numerous applications. We survey mostly new and also some old results on Macaulay posets, where the intention is to present them as pieces of a general theory. In

Cai, 1997

Let an edge cut partition the vertex set of an n-dimensional quadratic grid with the side length ... more Let an edge cut partition the vertex set of an n-dimensional quadratic grid with the side length a into k subsets A 1 , ..., A k with ||A i | − |A j || ≤ 1. We consider the problem of determining the minimal size c(n, k, a) of such a cut and present its asymptotic c(n, k, a) ∼ na n−1 n √ k as a, k → ∞ and k/a n → 0. The same asymptotic holds for partitioning of the n-dimensional torus. We present also some heuristics, which provide better partitioning for n = 2 and small k.

Theoretical Computer Science, 2003

We consider an edge-isoperimetric problem (EIP) on the cartesian powers of graphs. One of our obj... more We consider an edge-isoperimetric problem (EIP) on the cartesian powers of graphs. One of our objectives is to extend the list of graphs for whose cartesian powers the lexicographic order provides nested solutions for the EIP. We present several new classes of such graphs that include as special cases all presently known graphs with this property. Our new results are applied to derive best possible edge-isoperimetric inequalities for the cartesian powers of arbitrary regular, resp. regular bipartite, graphs with a high density.

This article is a study of the solution set of a discrete isoperimetric problem.

Cocoon, 1999

Let an edge cut partition the vertex set of a graph into k disjoint subsets A1, . . . , A k with ... more Let an edge cut partition the vertex set of a graph into k disjoint subsets A1, . . . , A k with ||Ai| − |Aj|| ≤ 1. We consider the problem of determining the minimal size of such a cut for a given graph. For this we introduce a new lower bound method which is based on the solution of an extremal set problem and present bounds for some graph classes based on Hamming graphs.

Discrete Mathematics, Jun 1, 2008

We introduce a new graph for all whose cartesian powers the vertex isoperimetric problem has nest... more We introduce a new graph for all whose cartesian powers the vertex isoperimetric problem has nested solutions. This is the fourth kind of graphs with this property besides the well-studied graphs like hypercubes, grids and tori. In contrast to the mentioned graphs, our graph is not bipartite. We present an exact solution to the vertex isoperimetric problem on our graph by introducing a new class of orders that unifies all known isoperimetric orders defined on the cartesian powers of graphs.

We present embeddings of generalized ladders as subgraphs into the hypercube. By embedding caterp... more We present embeddings of generalized ladders as subgraphs into the hypercube. By embedding caterpillars into ladders, we obtain embeddings of caterpillars into the hypercube. In this way we obtain almost all known results concerning the embeddings of caterpillars into the hypercube. In addition we construct embeddings for some new types of caterpillars.

This paper is a survey on discrete isoperimetric type problems. We present here as some known fac... more This paper is a survey on discrete isoperimetric type problems. We present here as some known facts about their solutions as well some new results and demonstrate a general techniques used in this area. The main attention is paid to the unit cube and cube like structures. Besides some applications of the isoperimetric approach are listed too.

Ann Comb, 2000

In this paper we introduce a new order on the set of n-dimensional tuples and prove that this ord... more In this paper we introduce a new order on the set of n-dimensional tuples and prove that this order preserves nestedness in the edge isoperimetric problem for the graph P n , de ned as the n th cartesian power of the well-known Petersen graph. The cutwidth and wirelength of P n are also derived. These results are then generalized for the cartesian product of P n and the m-dimensional binary hypercube.

The subject of this paper are grid re nements arising in nite element methods and their embedding... more The subject of this paper are grid re nements arising in nite element methods and their embedding into grids. We concentrate on embeddings with dilation 1 and construct for a class of grid re nements their embeddings with optimal load. For general cases we show that the problem to determine the minimal load is NP-complete.