, Volume 11, Issue 4, pp 299-314

Edge-isoperimetric inequalities in the grid

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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 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).