New results in t-tone coloring of graphs [article]

Daniel W. Cranston, Jaehoon Kim, William B. Kinnersley
2011 arXiv   pre-print
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 τ_t(G), is the minimum k such that G has a t-tone k-coloring. Bickle and Phillips showed that always τ_2(G) < [Δ(G)]^2 + Δ(G), but conjectured that in fact τ_2(G) < 2Δ(G) + 2; we confirm this conjecture when Δ(G) < 3 and also show that always τ_2(G) <(2 + √(2))Δ(G). For general t we prove that τ_t(G) <
more » ... ^2+t)Δ(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 √(Δ(T))<τ_t(T) < c_2√(Δ(T)).
arXiv:1108.4751v1 fatcat:r4qamkdqzvhpfaijp2zahne2x4