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$G$，能否以高概率通过其$k$-牌（即大小为$k$的所有诱导（着色）子图集合）重构其顶点在$r$种颜色下的均匀随机着色？本文研究了格点图与随机图中着色的重构问题。近期，Narayanan和Yap证明：对于$d=2$情况，若$k\geq\sqrt{2\log_2 n}+\frac{3}{4}$，则$d$维$n$-格（$n\times n$网格）顶点的随机着色可被其所有$k$子网格（$k\times k$网格）的牌高概率重构；若$k<\sqrt{2\log_2 n}-\frac{1}{4}$，则不可重构。我们证明这一"两点集中"结果对确定整体着色的最小子网格尺寸在任意维度$d\geq 2$时均成立。此外，我们证明随机图$G(n,1/2)$顶点的均匀随机$r$着色：当$k\geq 2\log_2 n+8$时可通过其完整$k$-牌高概率重构，当$k\leq\sqrt{2\log_2 n}$时则不可重构。进一步地，我们表明随机图的颜色重构算法可经修改用于图重构：当$k\geq 2\log_2 n+11$时，$G(n,1/2)$可通过其完整$k$-牌高概率重构（而若$k\leq 2\sqrt{\log_2 n}$，则高概率不可重构）。这显著改进了当前已知的、可高概率重构随机图所需牌中子图最小尺寸的上界。