Summary: \(T\)-colorings arose form the channel assignment problem in communications. Given a finite set \(T\) of non-negative integers, a \(T\)-coloring of a simple graph \(G\) is an assignment of a non-negative integer (channel) on every vertex of \(G\), such that the difference of channels of two adjacent vertices does not fall in \(T\). The \(T\)-span of \(G\), denoted by \(\text{sp}_T(G)\), is the minimum span among all possible \(T\)-colorings of \(G\), where the span of a \(T\)-coloring is the difference between the largest and smallest channels used. This article applies \(T\)-graphs to explore the sets \(T\) that belong to these two collections: \({\mathcal G}=\{T:\) greedy (or first-fit) algorithm provides \(T\)-spans for all complete graphs}, and \({\mathcal E}=\{T:\text{sp}_T(G)=\text{sp}_T(K_{\chi(G)})\) for all graphs \(G\), where \(\chi\) is the chromatic number}. Given \(T\), its \(T\)-graph has the vertex set of all non-negative integers so that two vertices are adjacent if their difference does not fall in \(T\). We show that for any \(a\in\mathbb{Z}^+\), the \(T\)-graph of \(aT\) is isomorphic to a disjoint product of \(a\) copies of the \(T\)-graph of \(T\), where \(aT\) is the set obtained by multiplying every element of \(T\) by \(a\). Based on this characterization, the following two results are attained. For any \(a\in\mathbb{Z}^+\), \(T\in{\mathcal E}\) if and only if \(aT\in{\mathcal E}\), and \(T\in{\mathcal G}\) if and only if \(aT\in{\mathcal G}\). The second fact has been proven by M. B. Cozzens and F. S. Roberts [J. Comb. Inf. Syst. Sci. 16, No. 4, 286-299 (1991; Zbl 0774.05037)] from a different approach. We will completely solve the family of sets \(T=\{0,s,s+1,\dots,l\}\) by providing a different proof of the fact \(T\in{\mathcal G}\) (B. A. Tesman, Ph.D. Dissertation, New Brunswick, 1989), and showing that \(T\in{\mathcal E}\) if and only if \(l\) is a multiple of \(s\). In addition, complete solutions for a more general family are obtained: for any \(a,s,l\in\mathbb{Z}^+\) with \(s\leq l\), \(T=\{0,as,a(s+1),a(s+2),\dots,al\}\cup A\in{\mathcal G}\), and \(T\in{\mathcal E}\) if and only if \(l=ms\) for some \(m\), where \(A\subseteq\{as+1,as+2,\dots,al-1\}\).