## Abstract

The grid graph is the graph on [*k*]^{ n }={0,...,*k*−1}^{ n } in which*x*=(*x*_{ i }) _{1} ^{ n } is joined to*y*=(*y*_{ i }) _{1} ^{ n } if for some*i* we have |*x*_{ i }*−y*_{ i }|=1 and*x*_{ j }=*y*_{ j } for all*j*≠*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 if*A*⊂[*k*]^{ n } satisfies*k*^{ n }/4≤|*A*|≤3*k*^{ n }/4 then there are at least*k*^{ n−1 } edges between*A* 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, for*r*=1,...,*k* we show that if*A*⊂[*k*]^{ n } satisfies |*A*|≤*r*^{ n } then the subgraph of [*k*]^{ n } induced by*A* has average degree at most 2*n*(1−1/*r*).

## AMS subject classification (1991)

05 C 35## Preview

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