What is "t-coloring"

In graph theory, a T-Coloring of a graph G = (V, E), given the set T of nonnegative integers containing 0, is a function c : V(G) → N that maps each vertex of G to a positive integer ( color) such that $(u,w) \in E(G) \Rightarrow \left | c(u) - c(w) \right | \notin T$. In simple words, the absolute value of the difference between two colors of adjacent vertices must not belong to fixed set T. The concept was introduced by William K. Hale. If T = {0} it reduces to common vertex coloring.

The complementary coloring of T-coloring c, denoted $\overline{c}$ is defined for each vertex v of G by
$\overline{c}(v) = s +1-c(v)$
where s is the largest color assigned to a vertex of G by the c function.