Edgeisoperimetric inequalities in the grid
 Béla Bollobás,
 Imre Leader
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The grid graph is the graph on [k]^{ n }={0,...,k−1}^{ n } in whichx=(x _{ i }) _{1} ^{ n } is joined toy=(y _{ i }) _{1} ^{ n } if for somei we have x _{ i } −y _{ i }=1 andx _{ j }=y _{ j } for allj≠i. In this paper we give a lower bound for the number of edges between a subset of [k]^{ n } of given cardinality and its complement. The bound we obtain is essentially best possible. In particular, we show that ifA⊂[k]^{ n } satisfiesk ^{ n }/4≤A≤3k ^{ n }/4 then there are at leastk ^{ n−1 } edges betweenA and its complement.
Our result is apparently the first example of an isoperimetric inequality for which the extremal sets do not form a nested family.
We also give a best possible upper bound for the number of edges spanned by a subset of [k]^{ n } of given cardinality. In particular, forr=1,...,k we show that ifA⊂[k]^{ n } satisfies A≤r ^{ n } then the subgraph of [k]^{ n } induced byA has average degree at most 2n(1−1/r).
 Bernstein, A. J.: Maximally connected arrays on thencube,SIAM J. Appl. Math. 15 (1967), 1485–1489.
 Bollobás, B.:Combinatorics, Cambridge University Press, 1986, xii+177 pp.
 Bollobás, B., andLeader, I.: Compressions and isoperimetric inequalities,J. Combinatorial Theory (A),56 (1991), 47–62.
 Clements, G. F.: Sets of lattice points which contain a maximal number of edges,Proc. Amer. Math. Soc. 27 (1971), 13–15.
 Frankl, P. The shifting technique in extremal set theory, inSurveys in Combinatorics 1987 (Whitehead, C., ed.), London Math. Soc. Lecture Note Series 123, Cambridge University Press, 1987, pp. 81–110.
 Harper, L. H.: Optimal assignments of numbers to vertices,SIAM J. Appl. Math. 12 (1964), 131–135.
 Hart, S. A note on the edges of thencube,Discrete Math. 14 (1976), 157–163.
 Kleitman, D. J., Krieger, M. M., andRothschild, B. L.: Configurations maximizing the number of pairs of Hammingadjacent lattice points.Studies in Appl. Math. 50 (1971), 115–119.
 Lindsey, J. H.: Assignment of numbers to vertices,Amer. Math. Monthly 71 (1964), 508–516.
 Wang, D.L., andWang, P.: Discrete isoperimetric problems,SIAM J. Appl. Math 32 (1977), 860–870.
 Title
 Edgeisoperimetric inequalities in the grid
 Journal

Combinatorica
Volume 11, Issue 4 , pp 299314
 Cover Date
 19911201
 DOI
 10.1007/BF01275667
 Print ISSN
 02099683
 Online ISSN
 14396912
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 05 C 35
 Industry Sectors
 Authors

 Béla Bollobás ^{(1)} ^{(2)}
 Imre Leader ^{(1)} ^{(2)}
 Author Affiliations

 1. Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, England
 2. Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana, USA