The grid graph is the graph on [k]n = {0 ..... k -1} n in which x = (xi) ~ is joined to y = (Yi)~ if for some i we have Ixi-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 C [k]n satisfies kn/4 < IAI < 3kn/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 C [k]n satisfies IAI < r n then the subgraph of [k]n induced by A has average degree at most 2n(1 -l/r). IntroductionIn this paper we are interested in edge-isoperimetric inequalities on graphs. Given a graph G and a natural number m, at most how many edges are spanned by a set of m vertices? Also, at least how many edges are there between a set of m vertices and its complement?To make these questions precise, let us introduce a small amount of notation. Given a graph G, define the edge-boundary Oe(A) and the edge-interior inte(A) of a set A C V(G) by Oe(A) = {xy E E(G): z E A, y r A}, Inte (A) = {xy E E(G) : x,y 9 A}.Then we wish to determine min{lOe(A)l: IAI = m} and max {lInte (A)I: IA] = m}.If G is a regular graph then these two problems happen to be equivalent. In order to state their result, define an order on .~n, the binary order , by letting a set S precede a set T if max(Sz~T) 9 T, in other words if the greatest element of {1,..., n} which is in one of S and T but not the other is actually in T. With this terminology, Harper, Lindsey, Bernstein and Hart proved that initial segments of the binary order are subsets of .2 n of minimum edge-boundary (and so maximum edge-interior). See [2, Ch.16] for a general discussion of this and related topics. Our aim in this paper is to give an answer to these questions for the grid[k]n = {0,..., k -1} n, where as usual x = (Xl,..., xn) is joined to y = (Yl,..., Yn)
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