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

A t-tone k-coloring of a graph G is a function f : V(G) -> (([k])(t)) such that vertical bar f (u) boolean AND f(v)vertical bar < d(u, v) for all u, v is an element of V(G) with u not equal v. We write [k] as shorthand for {1, ..., k} and denote by (([k])(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) <= 3, then tau(2) (G) <= 8. In this paper, we consider 3-tone coloring of graphs G with Delta(G) <= 3. The previous best result was that tau(3) (G) <= 36; here we show that tau(3) (G) <= 21.