Timetabling problems are, in practice, complex due to the many conditions which are imposed, and they cannot therefore readily be expressed as formal problems. A formal representation of some aspects is, however, possible; for example, the basic problem of allocating students to classes can be expressed as that of colouring the vertices of a graph so that no two adjacent vertices have the same colour. Thus, consideration of timetabling problems involves both direct heuristic methods and the solution of appropriate graph theoretic models. This thesis, therefore, commences by a study of relevant graph theory problems and, in particular, develops algorithms to determine the complete subgraph of maximum degree in a given graph and also that minimal set of edges or vertices which will disconnect a graph. The technique of removing vertices from a graph G, starting with those of the lowest degree, is proposed, and it is shown that this can be used to find an upper bound to the chromatic number of the graph G, or to give a subgraph containing a critical chromatic subgraph of G. Methods are established which enable the chromatic number of certain critical chromatic graphs to be determined, and a conjecture concerning the nature of such graphs is proposed. A further concept introduced is that of splitting or sub-dividing a vertex into two or more vertices; this has relevance in timetabling problems and appropriate algorithms for this process are developed. Finally, algorithms are derived, using these results as appropriate for several timetabling problems. The particular timetables studied include those for the allocation of lecturers and classes in a University department, both when the classes remain the same throughout the session, and when they consist of a number of separate courses given by different lecturers. The timetable is also obtained for the optimal arrangement of courses in a Faculty where students are able to choose the subjects studied. All these problems are of wider application.