New results in $t$-tone coloring of graphs

A $t$-tone $k$-coloring of $G$ assigns to each vertex of $G$ a set of $t$ colors from $\{1,..., k\}$ so that vertices at distance $d$ share fewer than $d$ common colors. The {\it $t$-tone chromatic number} of $G$, denoted $τ_t(G)$, is the minimum $k$ such that $G$ has a $t$-tone $k$-coloring. Bickle and Phillips showed that always $τ_2(G) \le [Δ(G)]^2 + Δ(G)$, but conjectured that in fact $τ_2(G) \le 2Δ(G) + 2$; we confirm this conjecture when $Δ(G) \le 3$ and also show that always $τ_2(G) \le \ceil{(2 + \sqrt{2})Δ(G)}$. For general $t$ we prove that $τ_t(G) \le (t^2+t)Δ(G)$. Finally, for each $t\ge 2$ we show that there exist constants $c_1$ and $c_2$ such that for every tree $T$ we have $c_1 \sqrt{Δ(T)} \le τ_t(T) \le c_2\sqrt{Δ(T)}$.

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