A local-global principle for vertex-isoperimetric problems

We consider the vertex-isoperimetric problem for cartesian powers of a graph G. A total order on the vertex set of G is called isoperimetric if the boundary of sets of a given size k is minimum for any initial segment of , and the ball around any initial segment is again an initial segment of . We prove a local-global principle with respect to the so-called simplicial order on Gn (see Section 2 for the definition). Namely, we show that the simplicial order n is isoperimetric for each n ≥ 1 iff it is so for n = 1, 2. Some structural properties of graphs that admit simplicial isoperimetric orderings are presented. We also discuss new relations between the vertex-isoperimetric problems and Macaulay posets.