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

Daniel W. Cranston, Hudson LaFayette

For a graph $G$ and $t,k\in {\mathbb{Z}}^{+}$ a \emph{$t$-tone $k$-coloring} of $G$ is a function $f:V(G)\to \left(\genfrac{}{}{0ex}{}{[k]}{t}\right)$ such that $|f(v)\cap f(w)|<d(v,w)$ for all distinc…