Distance graphs and the T-coloring problem

Abstract

The T-coloring problem is, given a graph G = ( V, E), a set T of nonnegative integers containing 0, and a ‘span’ bound s ⩾ 0, to compute an integer coloring f of the vertices of G such that | f( ν) − f( w)| ∉ T ∀ νw ∈ E and max f − min f ⩽ s. This problem arises in the planning of channel assignments for broadcast networks. When restricted to complete graphs, the T-coloring problem boils down to a number problem which can be solved efficiently for many types of sets T. The paper presents results indicating that this is not the case if the set T is arbitrary. To these ends, the class of distance graphs is introduced, which consists of all graphs G : G ≅ G( A) for some (finite) set of positive integers A, where G( A) is defined to be the graph with vertex set A and the edges ab: | a − b| ∈ A. Exploiting an equivalence between the complete graph T-coloring problem and the distance graph clique problem, it is shown that the complete graph T-coloring problem is NP-complete in the strong sense. Furthermore, one also obtains the NP-hardness of the corresponding c-approximation problem. Also discussed is the distance graph recognition problem which is an interesting topic in its own right. In particular, it is shown that ordered distance graphs can be recognized in polynomial time using linear programming techniques.