Greedy Graph Colouring is a Misleading Heuristic

Let G = (V,E) be a (finite, simple, undirected) graph with vertex set V and edge set E. A clique in G is a set of vertices, each of which is adjacent to every other vertex in this set. The maximum clique problem is to find a largest such set in a given graph; this is NP-hard [GJ90]. We denote the size of a maximum clique by ω. A colouring of G is an assignment of vertices to colours such that no two adjacent vertices are given the same colour. Determining the minimum number of colours χ required to colour a graph is also NP-hard [GJ90], but greedy algorithms may produce a (non-optimal) colouring in polynomial time. There are two “obvious” quadratic greedy colouring algorithms: one could give each vertex in turn the first available colour. Alternatively, for each colour in turn, one could try to give that colour to each vertex in turn. In fact these two algorithms produce the same result. Any colouring of a graph require at least ω colours (each vertex in a clique must be given a different colour). Thus a greedy colouring provides the bound part of a branch and bound algorithm for the maximum clique problem. We illustrate these concepts in Figure 1, and refer to papers by Tomita for algorithms [TK07, TSH10].