New results in t-tone coloring of graphs

Abstract

A t-tone k-coloring of G assigns to each vertex of G a set of t colors from {1, ... , k} so that vertices at distance d share fewer than d common colors. The t-tone chromatic number of G, denoted tau(t)(G), is the minimum k such that G has a t-tone k-coloring. Bickle and Phillips showed that always tau(2)(G) <= [Delta(G)](2) + Delta(G), but conjectured that in fact tau(2)(G) <= 2 Delta(G) + 2; we confirm this conjecture when Delta(G) <= 3 and also show that always tau(2)(G) <= [(2 + root 2)Delta(G)]. For general t we prove that tau(t)(G) <= (t(2) + t)Delta(G). Finally, for each t >= 2 we show that there exist constants c(1) and c(2) such that for every tree T we have c(1)root Delta(T) <= tau(t)(T) <= c(2)root Delta(T).