T-graphs and the channel assignment problem

Abstract T -colorings arose from 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 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: G = { T : greedy (or first-fit) algorithm provides T -spans for all complete graphs}, and E = { T : spt ( G ) = sp T ( K χ ( G ) ) for all graphs G , where χ 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 ∈ 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 ∈ Z + , T ∈ E if and only if aT ∈ E , and T ∈ G if and only if aT ∈ G . The second fact has been proven by Cozzens and Roberts (1991) from a different approach. We will completely solve the family of sets T = {0, s , s + 1,…, l } by providing a different proof of the fact T ∈ G (Tesman, 1989), and showing that T ∈ 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 ∈ Z + with s ⩽ l , T = {0, as , a ( s + 1), a ( s + 2),…, al } ∪ A ∈ E , and T ϵ E if and only if l = ms for some m , where A ⊆{ as + 1, as + 2, …, al - 1}.