Independent Sets in Graph Powers are Almost Contained in Juntas
Geometric and Functional Analysis(2008)
摘要
. Let G = ( V , E ) be a simple undirected graph. Define G n , the n -th power of G , as the graph on the vertex set V n in which two vertices ( u 1 , . . . , u n ) and ( v 1 , . . . , v n ) are adjacent if and only if u i is adjacent to v i in G for every i . We give a characterization of all independent sets in such graphs whenever G is connected and non-bipartite. Consider the stationary measure of the simple random walk on G n . We show that every independent set is almost contained with respect to this measure in a junta, a cylinder of constant co-dimension. Moreover we show that the projection of that junta defines a nearly independent set, i.e. it spans few edges (this also guarantees that it is not trivially the entire vertex-set). Our approach is based on an analog of Fourier analysis for product spaces combined with spectral techniques and on a powerful invariance principle presented in [MoOO]. This principle has already been shown in [DiMR] to imply that independent sets in such graph products have an influential coordinate. In this work we prove that in fact there is a set of few coordinates and a junta on them that capture the independent set almost completely.
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关键词
independent sets,intersecting families,product graphs,discrete Fourier analysis
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