Edge-isoperimetric inequalities in the grid

Abstract

The grid graph is the graph on [k]n={0,...,k−1}n in whichx=(x i ) n1 is joined toy=(y i ) n1 if for somei we have |x i −y i |=1 andx j =y j for allji. 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).

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Research partially supported by NSF Grant DMS-8806097

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Bollobás, B., Leader, I. Edge-isoperimetric inequalities in the grid. Combinatorica 11, 299–314 (1991). https://doi.org/10.1007/BF01275667

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AMS subject classification (1991)

  • 05 C 35