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.

翻译：给定图$G$，何时能够从它的$k$-副本（即所有大小为$k$的导出（染色）子图构成的集合）中，以高概率重构其顶点在$r$种颜色上的均匀随机染色？在本文中，我们重构了格点图与随机图的随机染色。最近，Narayanan和Yap证明了：对于$d=2$，当$k\geq\sqrt{2\log_2 n}+\frac{3}{4}$时，$d$维$n$-格点（$n\times n$网格）顶点的随机染色以高概率可从其所有$k$-子网格（$k\times k$网格）的副本中重构；当$k<\sqrt{2\log_2 n}-\frac{1}{4}$时则不可重构。我们证明，在任意维数$d\geq 2$中，关于决定整个染色的最小子网格尺寸的相同“两点集中”结果仍然成立。此外，我们还证明了：对于随机图$G(n,1/2)$顶点的均匀随机$r$-染色，当$k\geq 2\log_2 n+8$时，以高概率可从其完整$k$-副本中重构；当$k\leq\sqrt{2\log_2 n}$时则不可重构。进一步地，我们展示了随机图的染色重构算法可被修改并用于图的重构：当$k\geq 2\log_2 n+11$时，以高概率可从其完整$k$-副本中重构$G(n,1/2)$（而当$k\leq 2\sqrt{\log_2 n}$时以高概率不可重构）。这显著改进了可用于以高概率重构随机图的副本中子图最小尺寸的已知上界。