Fred S. Roberts

T -colorings of graphs: recent results and open problems

Abstract Suppose G is a graph and T is a set of nonnegative integers. A T -coloring of G is an assignment of a positive integer ƒ( x ) to each vertex x of G so that if x and y are joined by an edge of G , then |ƒ(x) - ƒ(y)ƒ| is not in T . T -colorings were introduced by Hale in connection with the channel assignment problem in communications. Here, the vertices of G are transmitters, an edge represents interference, ƒ( x ) is a television or radio channel assigned to x , and T is a set of disallowed separations for channels assigned to interfering transmitters. One seeks to find a T -coloring which minimizes either the number of different channels ƒ( x ) used or the distance between the smallest and largest channel. This paper surveys the results and mentions open problems concerned with T -colorings and their variations and generalizations.

On the Order Fill Rate in a Multi-Item, Base-Stock Inventory System

A customer order to a multi-item inventory system typically consists of several different items in different amounts. The probability of satisfying an arbitrary demand within a prespecified time window, termed the order fill rate, is an important measure of customer satisfaction in industry. This measure, however, has received little attention in the inventory literature, partly because its evaluation is considered a hard problem. In this paper, we study this performance measure for a base-stock system in which the demand process forms a multivariate compound Poisson process and the replenishment leadtimes are constant. We show that the order fill rate can be computed through a series of convolutions of one-dimensional compound Poisson distributions and the batch-size distributions. This procedure makes the exact calculation faster and much more tractable. We also develop simpler bounds to estimate the order fill rate. These bounds require only partial order-based information or merely the item-based information. Finally, we investigate the impact of the standard independent demand assumption when the demand is actually correlated across items.

On the possible merging functions

Abstract In this paper, we study merging functions, functions which combine individual judgements into a merged or aggregate or consensus judgement. In particular, we study such functions under several simple axioms, symmetry, linear homogeneity, and agreement (which says that if all individuals agree, the merged judgement agrees with those of all of the individuals). We show that under one or more of these assumptions, the possible merging procedures are very few if we want certain statements involving the merged functions to be meaningful in the precise sense used in the theory of measurement, and that in many cases the arithmetic mean or the geometric mean are the only possible merging functions. The results are applied to group consensus problems, to performance analysis of alternative new technologies or of students or job applicants, and to the development of measures of price level.

On scientific laws without dimensional constants

A foundational paper of Lute [S] shows that the general form of a “scientific law” is greatly restricted by knowledge of the “admissible transformations” of the dependent and independent variables, transformations such as that from grams to pounds or inches to meters. The restrictions are discovered by formulating a functional equation from knowledge of the admissible transformations. Lute’s basic approach has been clarified and extended by Lute [6, 71, Rozeboom [15, 161, Osborne [12], and Roberts and Rosenbaum [ 141. A fundamental assumption in all the results which have been obtained so far is that the admissible transformations can be applied independently to all of the independent variables. In this paper we modify this assumption, and discover that in this situation, knowledge of the admissible transformations does not always restrict the form of the scientific law as greatly as in the cases previously studied. Specifically, suppose x1, x2 ,..., x, + 1 are n + 1 variables, z is the set of admissible transformations for the ith variable, i= 1, 2,..., n + 1, and x,+ , is some unknown function u(x,, x2,..., x,). The problem is to find the general form of the function u knowing the sets 5, i.e., to find the general form of the “scientific law”

The median procedure on median graphs

Abstract A median of a profile π = (x1, …, xk) of vertices of a finite connected graph G is a vertex x for which ∑ki = 1 d(x, xi) is minimum, where d is the usual geodesic distance on G. The function Med whose domain is the set of all profiles and is given by Med(π) = {x: x is a median of π} is called the median procedure on G. In this paper, the median procedure is characterized for median graphs and cube-free median graphs.

A characterization of clique graphs

Abstract In a recent paper [3], Hamelink obtains an interesting sufficient condition for a graph to be a clique graph. In this paper, we give related conditions which are necessary as well as sufficient. As an application of our result we show that Hamelinks condition is also necessary in certain special cases and that here it can be greatly simplified. As another application, we derive certain theorems useful in practice in reducing the question of whether a given graph is a clique graph to whether certain smaller or simpler graphs are.

Computing the boxicity of a graph by covering its complement by cointerval graphs

Abstract If F is a family of sets, its intersection graph has the sets in F as vertices and an edge between two sets if and only if they overlap. This paper investigates the concept of boxicity of a graph G, the smallest n such that G is the intersection graph of boxes in Euclidean n-space. The boxicity, b(G), was introduced by Roberts in 1969 and has since been studied by Cohen, Gabai, and Trotter. The concept has applications to niche overlap (competition) in ecology and to problems of fleet maintenance in operations research. These applications will be described briefly. While the problem of computing boxicity is in general a difficult problem (it is NP-complete), this paper develops techniques for computing boxicity which give useful bounds. They are based on the simple observation that b(G)≤k if and only if there is an edge covering of G by spanning subgraphs of G , each of which is a cointerval graph, the complement of an interval graph (a graph of boxicity ≤1.).

On the compatibility between a graph and a simple order

Abstract Following a definition of Goodman [2], we define a notion of compatibility between an (unoriented) graph and a simple (linear) order. The notion of indifference graph, introduced in [6] for finite graphs, is extended to graphs of arbitrary cardinalities; and the graphs compatible with some simple order are characterized as precisely the indifference graphs. The uniqueness of the compatible simple order is investigated, and it is shown that there is “essentially” only one such for each indifference graph. A definition of compatibility between oriented graphs and simple orders is also introduced and the oriented graphs compatible with some simple order are characterized as the semiorders of Luce [5] and Scott and Suppes [7]. It is proved that there is essentially only one simple order compatible with each semiorder. Finally, the compatibility results are applied to solve the psychologically-motivated problem of representing a graph (oriented graph) by “just noticeable difference” intervals on the real line. In the work on infinite graphs, the Axiom of Choice is freely (and tacitly) assumed.

AXIOMATIC THERMODYNAMICS AND EXTENSIVE MEASUREMENT

Abstract : The paper applies set theory to various phases of thermodynamics and measurement systems.

Post-Secondary Education

Since 1970 the number of U.S. college and university students choosing to major in mathematics has declined sharply. There are several sources of data describing this decline and many conjectures about the causes. The situation is summarized in two sections of the paper below.