Edge-isoperimetric inequalities in the grid

Abstract

The grid graph is the graph on [k]n={0,..., k-1}n in which x=(xi)1n is joined to y=(yi)1n if for some i we have |xi-yi|=1 and xj=yj 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 kn/4≤|A|≤3 kn/4 then there are at least kn-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|≤rn then the subgraph of [k]n induced by A has average degree at most 2 n(1-1/r). © 1991 Akadémiai Kiadó.

Publication Title

Combinatorica

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