The grid graph is the graph on [k]n={0,...,k−1}n in whichx=(xi)1n is joined toy=(yi)1n if for somei we have |xi−yi|=1 andxj=yj 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 satisfieskn/4≤|A|≤3kn/4 then there are at leastkn−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|≤rn then the subgraph of [k]n induced byA has average degree at most 2n(1−1/r).
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