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 .^[1] 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.^[2] If T = {0} 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.^[1]

Proposition. .^[3]

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:

^[4]

Some bounds of the T-span are given below:

• For every k-chromatic graph G with clique of size and every finite set T of nonnegative integers containing 0,
• For every graph G and every finite set T of nonnegative integers containing 0 whose largest element is r, ^[5]
• For every graph G and every finite set T of nonnegative integers containing 0 whose cardinality is t, ^[5]

• Graph coloring