论文标题
无限方格网格的完美色素：覆盖物和双颜色
Perfect colorings of the infinite square grid: coverings and twin colors
论文作者
论文摘要
图$ g $的完美着色（等效概念是公平的分区和分区设计）是从一组顶点到某些有限的（颜色）的函数$ f $，以使每个颜色的节点$ i $都具有$ s（i，j）$ j $ j $ j $ j $ of $ j $，其中$ s（i，j）$是coldrix $ s $ s $ s $ s sporitix。如果$ s $是一组颜色上的一些简单图$ t $的邻接矩阵，则$ f $称为封面$ t $ cover Graph $ g $的覆盖。我们通过无限的方格网格来表征所有覆盖物，证明每种颜色都是轨道（即，在某些图形自动形态的作用下对应于轨道分区），或具有双颜色（即两种颜色，这样统一使它们保持完美的色彩）。双颜色的情况分别分类。 关键字：完美着色，公平分区，分区设计，方形网格，矩形网格，墙纸组，双色，图形覆盖
A perfect coloring (equivalent concepts are equitable partition and partition design) of a graph $G$ is a function $f$ from the set of vertices onto some finite 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 the set of colors, then $f$ is called a covering of the target graph $T$ by the cover graph $G$. We characterize all coverings by the infinite square grid, proving that every such coloring is either orbit (that is, corresponds to the orbit partition under the action of some group of graph automorphisms) or has twin colors (that is, two colors such that unifying them keeps the coloring perfect). The case of twin colors is separately classified. Keywords: perfect coloring, equitable partition, partition design, square grid, rectangular grid, wallpaper group, twin colors, graph covering
