Reconstruction of colours

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 ($n\times n$ grid) is reconstructibe from its deck of all $k$-subgrids ($k\times k$ grids) if $k\geq\sqrt{2\log_2 n}+\frac{3}{4}$ and is not reconstructible if $k<\sqrt{2\log_2 n}-\frac{1}{4}$. 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 $d\geq 2$. 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 $k\geq 2\log_2 n+8$ and is not reconstructible if $k\leq\sqrt{2\log_2 n}$. 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 $k\geq 2\log_2 n+11$ (while it is not reconstructible with high probability if $k\leq 2\sqrt{\log_2 n}$). 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.