The impact of the DIMACS special year on Graph Theory

J. Friedman, P. Tetali, and R. Thomas (chair)
September 24, 1997

1. Introduction

The committee contacted all participants in the 1991-1992 DIMACS Special Year on Graph Theory, and asked them to evaluate its impact, given the perspective of several years. The response was overwhelmingly positive. All respondents reported favorably on the impact the Special Year has had on their research and scientific and scholarly development. While some individuals refered to a stimulating atmosphere and valuable interactions they had during their visits, most of the responses were very specific, pointing to concrete results and journal articles that resulted from their DIMACS visits.

It would not be practical to describe all of these results in the report. Instead, we will mention only those that this committee felt were of special importance for the development of the field. The list of references, however, presents a complete list of articles supplied by the participants. Those works were either obtained during or inspired by the activities of the Special Year.

2. Workshops

Four workshops were organized during the Special Year.

Each workshop was attended by 60 to 80 scientists and graduate students. The topics covered a large spectrum of problems of current research interest, ranging from theoretical to applied to practical implementation issues.

3. Post-docs and visitors

The post-docs at the special year were Sandy Irani, Laszlo Pyber, Gabor Tardos, Prasad Tetali and Dirk Vertigan. Irani is an Assistant Professor of Information and Computer Science at the University of California, Irvine, Pyber and Tardos hold research positions at the Hungarian Academy of Sciences in Budapest, Tetali is an Assistant Professor of Mathematics at the Georgia Institute of Technology and Vertigan is an Assistant Professor of Mathematics at the Louisiana State University. Tetali reports that the most valuable aspect of the Special Year were the workshops; on the whole he is indebted to DIMACS for facilitating his transition from graduate school to a tenure-track appointment.

4. New results

As we already mentioned in the Introduction, many results were obtained during or immediately following the Special Year. Here we mention a select few.

4.1 Piercing convex sets

Noga Alon and Daniel Kleitman completely solved a problem raised by Hadwiger and Debrunner about fourty years ago. A family of sets in Rd has the (p, q)-property if among any p members of the family some q have a nonempty intersection. The Hadwiger and Debrunner's problem was whether or not for every p, q, d there exists an integer c such that every family of compact convex subsets of Rd having the (p, q)-property can be split into c or fewer families, each having nonempty intersection. Alon and Kleitman answered the question in the affirmative by using a fractional version of Helly's theorem, linear programming duality, and a result on epsilon-nets of convex sets.

4.2 Arrangeability, Ramsey numbers and game chromatic number

This section clearly demonstrates the benefits of a timely exchange of scientific ideas. At the Special Year workshop on Planar Graphs, G.~Chen presented his results obtained with R. Schelp on the linearity of Ramsey numbers for planar graphs. They used a concept they called ``arrangeability" of a graph, breaking their proof into two steps. First they showed that graphs of bounded arrangeability have linear Ramsey numbers, and then they demonstrated that every planar graph has arrangeability at most 761. Tom Trotter and Hal Kierstead were in the audience, and shortly thereafter realized that they can use arrangeability for a problem they were interested in: bounding the so-called game chromatic number of a graph. Indeed, they were able to show that every graph of bounded arrangeability has bounded game chromatic number, and hence, by the result of Chen and Schelp it follows that planar graphs have bounded game chromatic number. Furthermore, Kierstead and Trotter were able to improve upon the original bound of Chen and Schelp. That raised the question which classes of graphs have bounded arrangeability. Later, Rodl and Thomas proved that graphs with no subdivision isomorphic to a fixed graph do form such a class.

4.3 Quadratic dynamical systems

This is a new research direction initiated by Y. Rabinovich, A. Sinclair and A. Widgerson during Sinclair's visit to DIMACS. Quadratic dynamical systems are a natural class of non-linear stochastic processes that are used to model various phenomena in the natural sciences, such as population genetics and the kinetic theory of ideal gases. Less classically, they also provide an appropriate framework for the study of genetic algorithms for combinatorial optimization. In contrast to linear systems, which are well understood, there is little general theory available for the quantitative analysis of quadratic systems. Inspired by the many computational applications of linear stochastic systems in recent years, Rabinovich, Sinclair and Widgerson set out to investigate the potential of their quadratic counterparts.

They established the abstract combinatorial framework of QDSs, which is essential to understanding their computational potential, and derived many of their fundamental properties, such as convergence to a fixed point and the existence of linear invariants. They also posed several intriguing open questions. This work appeared at the IEEE FOCS in 1992 [57]. It has led to further work by several authors (see, e.g., [55, 12, 34]) and remains an active topic of research. One interesting more recent related development which they have been involved in is a quantitative analysis of crossover operators in population genetics [56].

4.4 Linkless embeddings of graphs in 3-space

Robertson, Seymour and Thomas studied linkless embeddings of graphs in 3-space. A piecewise-linear embedding of a graph in 3-space is called flat if every circuit of the graph bounds a disk disjoint from the rest of the graph. They have shown that:

4.5 Influence of variables on Boolean functions

Another significant result that was the outcome of the special year is [13], which since then led to very important results in random graphs and random structures in general. In the symmetric, special case of Bernoulli measure on the product space (which was an earlier result of Kahn-Kalai-Linial, FOCS '88), this result states that for every monotone Boolean function on {0, 1}n there exists a variable whose influence is at least (log n)/n, up to an absolute (multiplicative) constant. (Influence of a variable is a measure of how much the function depends on that variable, when the rest of the variables are assigned randomly.)

The work in [13] was fundamental in the work of Friedgut-Kalai [20] which, resolving a conjecture of Nati Linial, established results of the type - every monotone graph property has a sharp threshold. The result in [13] also had applications in learning theory and complexity theory (see e.g. [14]. More recently, Bourgain further generalized such results to monotone Boolean functions (on product spaces) which are not necessarily graph properties.

4.6 Hadwiger's conjecture for K6-free graphs

Robertson, Seymour and Thomas managed to settle the next open case of Hadwiger's conjecture. They have shown that every loopless graph which cannot be contracted onto the complete graph on six vertices is five-colorable. (They were awarded the D. Ray Fulkerson prize for this work).

5 Criticisms and suggestions for future improvements

Unlike the previous special year, which was the first of many special years at DIMACS and suffered the birth pains of any new enterprise, the 1991/92 Special Year went rather smoothly. Participants found DIMACS staff courteous and helpful, and facilities, including computer equipment, more than adequate. The only criticism the committee has heard was that the mathematics faculty at Rutgers were too busy to interact with on a regular basis.


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