Title:

A graph theory model for the computer solution of university timetables and related problems

The work described in this thesis is concerned with four main fields of investigation, three concerned with the problems of a university administration in producing timetables, and one concerned with the theory of graphs which provides a convenient mathematical model of a university coursestudent structure. A university administration's timetable problems may be classified under these headings: 1. the production of examination timetables, 2. the assignment of students to classes, and 3. the production of classteacherroom timetables. These three problems are a class of the general combinatorial problem and thus simple enumeration will, in theory, provide a solution. This thesis describes and evaluates several algorithmic methods of solution and several heuristic approaches to reduce the combinatorial difficulties of the problems. Although heuristic methods do not guarantee the finding of an optimal solution, or, in some cases, any solution at all, the success of particular heuristics is demonstrated an actual coursestudent data. A new algorithmic method is proposed for the construction of classteacherroom timetables. The feasibility of this method is demonstrated with a nontrivial example based on a game. The thesis concludes with an investigation of the theory of graphs, the mathematical model used in previous work. Upper and lower bounds for the chromatic number of a graph are developed and procedures for reducing the size of the problem are constructed and discussed. An algorithm for finding all the complete subgraphs of a graph is developed as an aid in determining the solution to parts of the timetable problem. This is then related to several theorems concerning the eigenvalues and eigenvectors of the matrices associated with graphs and their meaning in the terms of the structure of these graphs. This leads readily to a bound, involving eigenvalues, for the size of the largest complete subgraph in any given graph. The graph theory section ends with a short note on the four colour problem.
