Efficient bounds on a branch and bound algorithm for graph colouration

Abstract

By a proper colouring of a simple graph, we mean an assignment of colours to its vertices with adjacent vertices receiving different colours. Applications of this graph‐theoretic modelling are numerous, especially in the areas of scheduling and timetabling. We shall refer to a proper colouration that uses a smallest possible number of colours as a minimum (proper) colouration. This number for the graph is simply its chromatic number, the determination of which is well known to be a ‘hard’ problem. In this paper, from the computational point of view of actually constructing a colouration, we examine a simple coalescence/expansion procedure that bases on the branch and bound technique improved by efficient bounding conditions. They include various tightened fathoming criteria as well as complete subgraph determination. For the computer implementation, we also look at the issues of branching rules and the decomposition of solution tree. With these efforts, empirical results show that minimum colourations can be constructed for moderate size graphs, with ‘good’ solutions computed for problems of up to about forty vertices.