Enrollment generators, clustering and chromatic numbers

In this abstract, we study the timetable conflict graphs produced by an artificial generator of student enrolments. We find correlations of the chromatic number of such graphs with their density and clustering coefficient. The work gives evidence that the clustering coefficient is a useful measure of a graph. Our motivations for this work were initial arose during studies of space planning. A task with large financial implications for Universities is the planning and management of teaching space. There is evidence that teaching space is currently poorly utilised (HEFCE 1999), and we are developing methods with the aim of improving this situation (Beyrouthy et al 2006, 2007c,b). However, significant barriers in such development are (i) the lack of realistic data instances, and (ii) the lack of good understanding of the nature of the timetabling problems that arise in practice, and for which the space provisions need to be targetted. For good space planning, we believe it is advantageous to be able to simulate many future scenarios, but this cannot be done well without the ability to create realistic scenarios, tailored to a particular institution. Such creation inevitably requires a good understanding of the structure of problems that arise, as well as a good ability to solve them close to optimality. A lot of work in the timetabling community has been directed at the solvers. In this project we are also studying the structures of the problems themselves. We reported some initial studies in Beyrouthy et al (2007a) of timetabling conflict graphs: vertices represent events, and edges correspond to conflicts between events preventing them taking place at the same time. We introduced their study from the perspective of the clustering coefficient, c, of the graphs. The clustering coefficient has achieved significant usage in the study of networks (Watts and Strogatz 1998). Roughly speaking, it measures the probability that for any node in the graph, any two of its neighbours are also neighbours of each other. Specifically, the clustering coefficient, ci of a node, i (of degree at least two) is the density of induced subgraph given by the set of nodes that are adjacent to i. That is, ci is the probability that