Some progress on $t$-tone coloring

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.