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Independent sets in graph powers are almost contained in juntas. (English) Zbl 1147.05058
Summary: 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 $$(u_{1},\dots, u_{n})$$ and $$(v_{1},\dots, v_{n})$$ 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 [E. Mossel, R. O’Donnell, O. Regev, J. E. Steif and B. Sudakov, “Noise stability of functions with low influences: invariances and optimality, Proc. 46th IEEE Symp. on Foundations of Computer Sciences, 21–30 (2005)]. This principle has already been shown in [I. Dinur, E. Mossel and O. Regev, Conditional hardness for approximate coloring, in: Proc. 38th ACM Symp. on Theory of Computing, STOC’06, ACM, New York, 344–353 (2006)] 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.

MSC:
 05D05 Extremal set theory
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