Isoperimetric Problems in Discrete Spaces

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.

[1]  L. H. Harper.  Optimal Assignments of Numbers to Vertices , 1964 .

[2]  John H. Lindsey,et al.  Assignment of Numbers to Vertices , 1964 .

[3]  G. F. Clements,et al.  A sequence of (±1)-determinants with large values , 1965 .

[4]  L. H. Harper.  Optimal numberings and isoperimetric problems on graphs , 1966 .

[5]  Kenneth Steiglitz,et al.  Encoding of analog signals for binary symmetric channels , 1966, IEEE Trans. Inf. Theory.

[6]  A. J. Bernstein,et al.  Maximally Connected Arrays on the n-Cube , 1967 .

[7]  B. Lindström,et al.  A combinatorial problem in the $k$-adic number system , 1967 .

[8]  Bernt Lindström,et al.  The optimal number of faces in cubical complexes , 1971 .

[9]  Daniel J. Kleitman,et al.  Configurations Maximizing the Number of Pairs of Hamming-Adjacent Lattice Points , 1971 .

[10]  G. F. Clements.  Sets of lattice points which contain a maximal number of edges , 1971 .

[11]  Gyula O. H. Katona,et al.  The Hamming-sphere has minimum boundary , 1975 .

[12]  Sergiu Hart,et al.  A note on the edges of the n-cube , 1976, Discret. Math..

[13]  Rudolf Ahlswede,et al.  Contributions to the geometry of hamming spaces , 1977, Discret. Math..

[14]  Da-Lun Wang,et al.  Discrete Isoperimetric Problems , 1977 .

[15]  Da-Lun Wang,et al.  Extremal Configurations on a Discrete Torus and a Generalization of the Generalized Macaulay Theorem , 1977 .

[16]  R. Osserman.  The isoperimetric inequality , 1978 .

[17]  D. J. Kleitman.  Surveys in Combinatorics: Extremal hypergraph problems , 1979 .

[18]  Zoltán Füredi,et al.  A short proof for a theorem of Harper about Hamming-spheres , 1981, Discret. Math..

[19]  Rudolf Ahlswede,et al.  Note on an extremal problem arising for unreliable networks in parallel computing , 1983, Discret. Math..

[20]  Victor K.-W. Wei,et al.  Odd and even hamming spheres also have minimum boundary , 1984, Discret. Math..

[21]  Peter Frankl,et al.  A new short proof for the Kruskal-Katona theorem , 1984, Discret. Math..

[22]  ON THE DISTRIBUTION OF THE NUMBER OF INTERIOR POINTS IN SUBSETS OF THE n-DIMENSIONAL UNIT CUBE , 1984 .

[23]  H. J. Tiersma.  A note on Hamming spheres , 1985, Discret. Math..

[24]  Victor K.-W. Wei,et al.  Addendum to "odd and even hamming spheres also have minimum boundary" , 1986, Discret. Math..

[25]  Zoltán Füredi,et al.  Families of finite sets with minimum shadows , 1986, Comb..

[26]  Daniel J. Kleitman.  On a problem of Yuzvinsky on separating the n-cube , 1986, Discret. Math..

[27]  Sergei L. Bezrukov.  ON THE CONSTRUCTION OF SOLUTIONS OF A DISCRETE ISOPERIMETRIC PROBLEM IN HAMMING SPACE , 1989 .

[28]  Peter Frankl.  A lower bound on the size of a complex generated by an antichain , 1989, Discret. Math..

[29]  Béla Bollobás,et al.  Isoperimetric Inequalities for Faces of the Cube and the Grid , 1990, Eur. J. Comb..

[30]  Béla Bollobás,et al.  An Isoperimetric Inequality on the Discrete Torus , 1990, SIAM J. Discret. Math..

[31]  Béla Bollobás,et al.  Edge-isoperimetric inequalities in the grid , 1991, Comb..

[32]  Béla Bollobás,et al.  Compressions and isoperimetric inequalities , 1990, J. Comb. Theory, Ser. A.

[33]  Béla Bollobás,et al.  Isoperimetric inequalities and fractional set systems , 1991, J. Comb. Theory, Ser. A.

[34]  Specification of all Solutions of the Discrete Isoperimetric Problem that Have a Critical Cardinality , 2002 .

[35]  Extremal Ideals of the Lattice of Multisets , 2002 .