An Upper Bound for the 3-Tone Chromatic Number of Graphs with Maximum Degree 3

Abstract

A t-tone k-coloring of a graph G is a function $f: V(G)\rightarrow {[k]\atopwithdelims ()t}$ such that $|f(u)\cap f(v)|<d(u,v)$ for all $u,v\in V(G)$ with $u\ne v$. We write [k] as shorthand for $\{1,\ldots ,k\}$ and denote by ${[k]\atopwithdelims ()t}$ the family of t-element subset of [k]. The t-tone chromatic number of G, denoted $\tau _t(G)$, is the minimum k such that G has a t-tone k-coloring. Cranston, Kim, and Kinnersley proved that if G is a graph with $\Delta (G)\le 3$, then $\tau _2(G)\le 8$. In this paper, we consider 3-tone coloring of graphs G with $\Delta (G)\le 3$. The previous best result was that $\tau _3(G)\le 36$; here we show that $\tau _3(G)\le 21$.