ON THE MOBILE RADIO FREQUENCY ASSIGNMENT PROBLEM AND THE TRAFFIC LIGHT PHASING PROBLEM

The problem of assigning frequencies to mobile radio telephones and the problem of assigning green light times to traffic streams at an intersection both deal with the assignment of a set S(x) to each vertex x of a graph G. The frequency assignment problem is concerned with set colorings: S(x) and S(y) are disjoint if x and y are joined by an edge. The traffic light phasing problem is concerned with set phasings: S(x) and S(y) are disjoint if x and y are not joined by an edge. This paper obtains some results on the existence and minimum size of set colorings and phasings under special restrictions on the sets S(x), for example, that each S(x) is a real interval or a set of n integers, or a set with at least a certain number of elements. Some of these results extend recent work on the n‐tuple chromatic number. Finally, the set phasing and coloring problems are related to the problem of assigning sets S(x) to vertices of a graph so that S(x) and S(y) overlap if and only if x and y are joined by an edge. Results about existence and minimum size of such set assignments are given.