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A perfect coloring (equivalent concepts are equitable partition and partitiondesign) of a graph $G$ is a function $f$ from the set of vertices onto somefinite set (of colors) such that every node of color $i$ has exactly $S(i,j)$neighbors of color $j$, where $S(i,j)$ are constants, forming the matrix $S$called quotient. If $S$ is an adjacency matrix of some simple graph $T$ on theset of colors, then $f$ is called a covering of the target graph $T$ by thecover graph $G$. We characterize all coverings by the infinite square grid,proving that every such coloring is either orbit (that is, corresponds to theorbit partition under the action of some group of graph automorphisms) or hastwin colors (that is, two colors such that unifying them keeps the coloringperfect). The case of twin colors is separately classified.  Keywords: perfect coloring, equitable partition, partition design, squaregrid, rectangular grid, wallpaper group, twin colors, graph covering

Perfect colorings of the infinite square grid: coverings and twin colors