In graph theory, a T-Coloring of a graph , given the set T of nonnegative integers containing 0, is a function that maps each vertex to a positive integer (color) such that if u and w are adjacent then . 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 = it reduces to common vertex coloring.
The T-chromatic number, is the minimum number of colors that can be used in a T-coloring of G.
The complementary coloring of T-coloring c, denoted is defined for each vertex v of G by
where s is the largest color assigned to a vertex of G by the c function.
Proof. Every T-coloring of G is also a vertex coloring of G, so Suppose that and Given a common vertex k-coloring function using the colors We define as
For every two adjacent vertices u and w of G,
so Therefore d is a T-coloring of G. Since d uses k colors, Consequently,
The span of a T-coloring c of G is defined as
The T-span is defined as:
Some bounds of the T-span are given below: