Optimum Allocation of Rooms in Timetables

\( X \)-proper edge colourings of bipartite graphs are defined. These colourings arise in timetables where rooms have to be assigned to courses. The objective is to minimize the number of different rooms in which each course must be taught. An optimum assignment is represented by a \( k \)-optimum edge colouring of a bipartite graph. Some necessary conditions for a \( k \)-optimum colouring are obtained, in terms of forbidden subgraphs. An algorithm based on removing these forbidden subgraphs to obtain improved colourings is described.