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).
 Title
 Edgeisoperimetric inequalities in the grid
 Journal

Combinatorica
Volume 11, Issue 4 , pp 299314
 Cover Date
 199112
 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