平面半正镶嵌的完美精确着色
Perfect precise colorings of plane semiregular tilings
AI总结：
本文研究平面半正镶嵌的完美精确着色问题，为$k\leq 6$的某些$k$价半正镶嵌族构造了满足对称群置换与顶点邻接颜色约束的着色方案。
摘要：
平面半正镶嵌$\mathcal{T}$的着色是指为$\mathcal{T}$的每个砖块分配唯一颜色。若$G$是$\mathcal{T}$的对称群，当$G$中每个元素在有限颜色集上诱导一个置换时，称该着色为完美的。若$\mathcal{T}$是$k$价的，则使用$k$种颜色对$\mathcal{T}$着色时，若共享同一顶点的任意两个砖块颜色不同，称其为精确的。本文中，我们获得了平面上$k\leq 6$的某些$k$价半正镶嵌族的完美精确着色。
英文摘要：
A coloring of a planar semiregular tiling $\mathcal{T}$ is an assignment of a unique color to each tile of $\mathcal{T}$. If $G$ is the symmetry group of $\mathcal{T}$, we say that the coloring is perfect if every element of $G$ induces a permutation on the finite set of colors. If $\mathcal{T}$ is $k$-valent, then a coloring of $\mathcal{T}$ with $k$ colors is said to be precise if no two tiles of $\mathcal{T}$ sharing the same vertex have the same color. In this work, we obtain perfect precise colorings of some families of $k$-valent semiregular tilings in the plane, where $k\leq 6$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
