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 $$c: V(G) \to \N$$ that maps each vertex to a positive integer (color) such that if u and w are adjacent then $$|c(u) - c(w)| \notin T$$. 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, $$\chi_{T}(G),$$ is the minimum number of colors that can be used in a T-coloring of G.

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.

Relation to Chromatic Number

 * Proposition. $$\chi_{T}(G)=\chi(G)$$.

Proof. Every T-coloring of G is also a vertex coloring of G, so $$\chi_{T}(G)\geq \chi(G).$$ Suppose that $$\chi(G)=k$$ and $$r=\max(T).$$ Given a common vertex k-coloring function $$c: V(G) \to \N$$ using the colors $$\{1, \ldots,k\}.$$ We define $$d: V(G) \to \N$$ as


 * $$d(v)=(r+1)c(v)$$

For every two adjacent vertices u and w of G,


 * $$|d(u) - d(w)| =| (r+1)c(u) - (r+1)c(w)| =(r+1) | c(u)-c(w)| \geq r +1 $$

so $$|d(u) - d(w)| \notin T.$$ Therefore d is a T-coloring of G. Since d uses k colors, $$\chi_{T}(G)\leq k =\chi(G).$$ Consequently, $$\chi_{T}(G)=\chi(G).$$

T-span
The span of a T-coloring c of G is defined as


 * $$sp_T(c) = \max_{u,w \in V(G)} |c(u) -c(w)|.$$

The T-span is defined as:


 * $$sp_T(G) = \min_c sp_T(c).$$

Some bounds of the T-span are given below:


 * For every k-chromatic graph G with clique of size $$\omega$$ and every finite set T of nonnegative integers containing 0, $$sp_T(K_{\omega}) \le sp_T(G) \le sp_T(K_k).$$


 * For every graph G and every finite set T of nonnegative integers containing 0 whose largest element is r, $$sp_T(G)\le (\chi(G)-1)(r+1).$$


 * For every graph G and every finite set T of nonnegative integers containing 0 whose cardinality is t, $$sp_T(G)\le (\chi(G)-1)t.$$