The t-tone chromatic number of classes of sparse graphs

For a graph G and t, k E Z(+) a t-tone k-coloring of G is a function f : V (G) ? ?([k]) t) such that |f(v) n f(w)| < d(v, w) for all distinct v, w E t V (G). The t-tone chromatic number of G, denoted T-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 T-2(G) exactly when mad(G) < 12/5 and bound T-2(G), up to a small additive constant, when G is outerplanar. We also determine T-t(Cn) exactly when t ? {3, 4, 5}.