The $t$-Tone Chromatic Number of Classes of Sparse Graphs

For a graph $G$ and $t,k\in\mathbb{Z}^+$ a \emph{$t$-tone $k$-coloring} of $G$ is a function $f:V(G)\rightarrow \binom{[k]}{t}$ such that $|f(v)\cap f(w)| < d(v,w)$ for all distinct $v,w \in V(G)$. The \emph{$t$-tone chromatic number} of $G$, denoted $\tau_t(G)$, is the minimum $k$ such that $G$ is $t$-tone $k$-colorable. For small values of $t$, we prove sharp or nearly sharp upper bounds on the $t$-tone chromatic number of various classes of sparse graphs. In particular, we determine $\tau_2(G)$ exactly when $\textrm{mad}(G) < 12/5$ and bound $\tau_2(G)$, up to a small additive constant, when $G$ is outerplanar. We also determine $\tau_t(C_n)$ exactly when $t\in\{3,4,5\}$.