A proper 2-tone $k$-coloring of a graph is a labeling of the vertices with elements from $\binom{ [k]}{ 2}$ 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 $\tau_2(G)$ is the smallest $k$ such that $G$ admits a proper 2-tone $k$ coloring. In this paper, we prove that w.h.p. for $p\ge Cn^{ -1/4}\ln^{ 9/4}n$, $\tau_2(G_{ n,p})=(2+o(1))\chi(G_{ n,p})$ where $\chi$ represents the ordinary chromatic number. For sparse random graphs with $p=c/n$, $c$ constant, we prove that $\tau_2(G_{ n,p}) = \lceil{ { \sqrt{ 8\Delta+1} +5}/{ 2}}\rceil$ where $\Delta$ represents the maximum degree. For the more general concept of $t$-tone coloring, we achieve similar results.
Categories: math.CO