A local-global principle for vertex-isoperimetric problems

Authors:
Sergei L. Bezrukov;Oriol Serra
Affiliations:
Department of Mathematics and Computer Science, University of Wisconsin-Superior, Superior, WI;Departament de Matemtica Aplicada i Telematica, modul C3, Campus Nord, Universitat Politecnica de Catalunya, Jordi Girona 1, 08034 Barcelona, Spain
Venue:
Discrete Mathematics - Kleitman and combinatorics: a celebration
Year:
2002

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Abstract

We consider the vertex-isoperimetric problem (VIP) 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 VIP and Macaulay posets.