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In graph theory, a T-Coloring of a graph G = {\displaystyle G=} , given the set T of nonnegative integers containing 0, is a function c : V → N {\displaystyle c:V\to \mathbb {N} } that maps each vertex to a positive integer such that if u and w are adjacent then | c − c | ∉ T {\displaystyle |c-c|\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 T-chromatic number, χ T , {\displaystyle \chi _{T},} is the minimum number of colors that can be used in a T-coloring of G.
The complementary coloring of T-coloring c, denoted c ¯ {\displaystyle {\overline {c}}} 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.