The t-Tone Chromatic Number of Random Graphs

A proper 2-tone k-coloring of a graph is a labeling of the vertices with elements from $${\binom{[k]}{2}}$$[k]2 such that adjacent vertices receive disjoint labels and vertices distance 2 apart receive distinct labels. The 2-tone chromatic number of a graph G, denoted τ2(G) is the smallest k such that G admits a proper 2-tone k coloring. In this paper, we prove that w.h.p. for $${p\geq Cn^{-1/4} {\rm ln}^{9/4}n, \tau_2(G_{n, p}) = (2 + o(1))\chi(G_{n, p})}$$p≥Cn-1/4ln9/4n,τ2(Gn,p)=(2+o(1))χ(Gn,p) where $${\chi}$$χ represents the ordinary chromatic number. For sparse random graphs with p = c/n, c constant, we prove that $${\tau_2(G_{n, p}) = \lceil{({\sqrt{8\Delta + 1} + 5})/{2}}}$$τ2(Gn,p)=⌈(8Δ+1+5)/2 where Δ represents the maximum degree. For the more general concept of t-tone coloring, we achieve similar results.