Locally biased partitions of [formula omitted

Abstract

Given a function f on the vertex set of some graph G, a scenery, let a simple random walk run over the graph and produce a sequence of values. Is it possible to, with high probability, reconstruct the scenery f from this random sequence?To show this is impossible for some graphs, Gross and Grupel, in Gross and Grupel (2018), call a function f:V→{0,1} on the vertex set of a graph G=(V,E)locallyp-biased if for each vertex v the fraction of neighbours on which f is 1 is exactly p. Clearly, two locally p-biased functions are indistinguishable based on their sceneries. Gross and Grupel construct locally p-biased functions on the hypercube {0,1}n and ask for what p∈[0,1] there exist locally p-biased functions on Zn and additionally how many there are. We fully answer this question by giving a complete characterization of these values of p. We show that locally p-biased functions exist for all p=c∕2n with c∈{0,…,2n} and that, in fact, there are uncountably many of them for every c∈{1,…,2n−1}. To this end, we construct uncountably many partitions of Zn into 2n parts such that every element of Zn has exactly one neighbour in each part. This additionally shows that not all sceneries on Zn can be reconstructed from a sequence of values attained on a simple random walk.