Abstract. We discuss four variants of the graph colouring problem, and present algorithms for solving them. The problems are k-Colourability, Max Ind k-COL, Max Val k-COL, and, finally, Max k-COL, which is the unweighted case of the Max k-Cut problem. The algorithms are based on the idea of partitioning the domain of the problems into disjoint subsets, and then considering all possible instances were the variables are restricted to values from these partitions. If a pair of variables have been restricted to different partitions, then the constraint between them is always satisfied since the only allowed constraint is disequality. 1

Bicliques of graphs have been studied extensively, partially motivated by the large number of applications. One of the main algorithmic interests is in designing algorithms to enumerate all maximal bicliques of a (bipartite) graph. Polynomial-time reductions have been used explicitly or implicitly to design polynomial delay algorithms to enumerate all maximal bicliques. Based on polynomial-time Turing reductions, various algorithmic problems on (maximal) bicliques can be studied by considering the related problem for (maximal) independent sets. In this line of research, we improve Prisner’s upper bound on the number of maximal bicliques [Combinatorica, 2000] and show that the maximum number of maximal bicliques in a graph on n vertices is exactly 3 n/3 (up to a polynomial factor). The main results of this paper are O(1.3642 n) time algorithms to compute the number of maximal independent sets and maximal bicliques in a graph.