## Compressions and isoperimetric inequalities.(English)Zbl 0731.05043

Let $$G=(V,E)$$ be a graph. For $$A\subset V$$ and $$y\in V$$, set $$D(A,y)=\inf \{d(x,y):$$ $$x\in A\}$$, where d is the usual graph metric. For $$t=0,1,2,...$$, $$A_{(t)}=\{y\in V:$$ d(A,y)$$\leq t\}$$ is the t-boundary of A and $$A_{(1)}=\partial A$$ is the boundary of A. $$Z^ n_+$$ is viewed as a graph on $$V=Z^ n_+=\{x\in Z^ n:$$ $$x_ i\geq 0$$ for all $$i\}$$ in which x is adjacent to y if for some i, $$| x_ i-y_ i| =1$$ and $$x_ j=y_ j$$ for all $$j\neq i$$. This is the infinite grid, the Cartesian product of infinite paths $$Z_+$$. $$[k]^ n$$ is the Cartesian product of n paths [k] on $$V=[k]=\{0,1,...,k-1\}$$ with i adjacent to i-1 for $$1\leq i\leq k-1$$. The simplicial order on $$Z^ n_+$$ is given by: $$x<y$$ if either $$\Sigma x_ i<\Sigma y_ i$$, or $$\Sigma x_ i=\Sigma y_ i$$ and for some j, $$x_ j>y_ j$$ and for all $$i<j$$, $$x_ i=y_ i$$. Only finite subsets A of $$Z^ n_+$$ are considered. For $$m=0,1,...$$, $$\partial^ n(m)$$ denotes the size of the boundary of the first m points of $$Z^ n_+$$ in the simplicial order. The restriction of the above order to $$[k]^ n$$ provides it with its simplicial order. Similar terminology is adopted for this. In the first part the authors prove $(1)\text{ for } finite\quad A\subset Z^ n_+,\quad | \partial A| \geq \partial^{(n)}(| A|),$
$(2)\text{ for } A\subset [k]^ n,\quad | \partial A| \geq \partial_ k^{(n)}(| A|).$ The first is a result due to D. L. Wang and P. Wang [Discrete isoperimetric problems, SIAM J. Appl. Math. 32, 860-870 (1977; Zbl 0362.05047)]. The proof technique employed is compression of a set. Roughly speaking, for a coordinate direction i, the i-compression of a set A replaces it by an equinumerous set $$C_ i(A)$$ which is ‘convex’ (‘regular’) without gaps in the i-direction, and ‘touches’ the i-th coordinate plane. The i-direction can be replaced by any vector direction.
For a graph G of diameter D and given $$\epsilon$$, $$0<\epsilon <1$$, let $$\alpha (G,\epsilon)=\min \{1-| A_{(\epsilon D)}| /| V|:\;A\subset V,\quad | A| /| V| \geq \}.$$ Then a family $$(G_ n)_ 1^{\infty}$$ of graphs is a normal Levy family if there are constants $$c_ 1,c_ 2>0$$ such that $$\alpha (G_ n,\epsilon)\leq c_ 1e^{-c_ 2\epsilon^ 2n}$$ for all n and $$\epsilon$$. In the second part, the authors extend inequality (2) to a finite product of arbitrary graphs and deduce therefrom that $$(G_ n)$$ is a normal Levy family with $$c_ 2=6D^ 2/(k^ 2-1)$$, for a connected graph G with diameter D and order k. This improves a result of N. Alon and V. D. Milman $$[\lambda_ 1$$, isoperimeric inequalities, and superconcentrators, J. Comp. Theory. Ser. B 38, 73-88 (1985; Zbl 0549.05051)].
Let $$X^{(r)}=\{1,2,...,n\}^ r$$ and let $${\mathcal A}$$ be a set system on $$X^{(r)}$$. The (lower) shadow of $${\mathcal A}$$ is $$\partial^-{\mathcal A}=\{B\subset X^{(r-1)}:\;B\subset A\text{ for some } A\in {\mathcal A}\}.$$ The colex order on $$X^{(r)}$$ is given by $$A<B$$ if max(A$$\Delta$$ B)$$\in B$$. Let $${\mathcal J}$$ denote the set of the first $$| {\mathcal A}|$$ elements in the colex order on $$X^{(r)}$$. In the final part, the authors give a direct, short proof of the following theorem of Kruskal and Katona using an extended notion of compression $(3)\quad | \partial^-{\mathcal A}| \geq | \partial^-{\mathcal J}|.$

### MSC:

 05C99 Graph theory 05C38 Paths and cycles 52B60 Isoperimetric problems for polytopes 05C12 Distance in graphs

### Citations:

Zbl 0362.05047; Zbl 0549.05051
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### References:

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