A $t$-tone coloring of a graph $G$ assigns to each vertex a set of $t$ colors such that any pair of vertices $u, v$ with distance $d$ can share at most $d-1$ colors. In this note, we prove several new results on $t$-tone coloring. For example we prove a new result for trees of large maximum degree, as well as some results for the cartesian power of a graph. We also make a conjecture about trees.

翻译：图的t-色调着色为每个顶点分配t种颜色的集合，使得任意距离为d的顶点对u、v至多共享d-1种颜色。本文证明了t-色调着色的若干新结果，例如针对最大度较大树的新结论，以及关于图笛卡尔幂的一些结果。同时提出了关于树的一个猜想。