Reconstruction of colours
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Given a graph G, when is it possible to reconstruct with high probability a uniformly random colouring of its vertices in r colours from its k-deck, i.e. a set of its induced (coloured) subgraphs of size k? In this paper, we reconstruct random colourings of lattices and random graphs. Recently, Narayanan and Yap proved that, for d=2, with high probability a random colouring of vertices of a d-dimensional n-lattice (n× n grid) is reconstructibe from its deck of all k-subgrids (k× k grids) if k≥√(2log_2 n)+3/4 and is not reconstructible if k<√(2log_2 n)-1/4. We prove that the same "two-point concentration" result for the minimum size of subgrids that determine the entire colouring holds true in any dimension d≥ 2. We also prove that with high probability a uniformly random r-colouring of the vertices of the random graph G(n,1/2) is reconstructible from its full k-deck if k≥ 2log_2 n+8 and is not reconstructible if k≤√(2log_2 n). We further show that the colour reconstruction algorithm for random graphs can be modified and used for graph reconstruction: we prove that with high probability G(n,1/2) is reconstructible from its full k-deck if k≥ 2log_2 n+11 (while it is not reconstructible with high probability if k≤ 2√(log_2 n)). This significantly improves the best known upper bound for the minimum size of subgraphs in a deck that can be used to reconstruct the random graph with high probability.
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