In graph theory, a domatic partition of a graph is a partition of into disjoint sets , ,..., such that each Vi is a dominating set for G. The figure on the right shows a domatic partition of a graph; here the dominating set consists of the yellow vertices, consists of the green vertices, and consists of the blue vertices. The domatic number is the maximum size of a domatic partition, that is, the maximum number of disjoint dominating sets. The graph in the figure has domatic number 3. It is easy to see that the domatic number is at least 3 because we have presented a domatic partition of size 3. To see that the domatic number is at most 3, we first review a simple upper bound. Upper bounds Let be the minimum degree of the graph . The domatic number of is at most . To see this, consider a vertex of degree . Let consist of and its neighbours. We know that (1) each dominating set must contain at least one vertex in (domination), and (2) each vertex in is contained in at most one dominating set (disjointness). Therefore, there are at most disjoint dominating sets. The graph in the figure has minimum degree , and therefore its domatic number is at most 3. Hence we have shown that its domatic number is exactly 3; the figure shows a maximum-size domatic partition. Lower bounds If there is no isolated vertex in the graph (that is, ≥ 1), then the domatic number is at least 2. To see this, note that (1) a weak 2-coloring is a domatic partition if there is no isolated vertex, and (2) any graph has a weak 2-coloring. Alternatively, (1) a maximal independent set is a dominating set, and (2) the complement of a maximal independent set is also a dominating set if there are no isolated vertices. The figure on the right shows a weak 2-coloring, which is also a domatic partition of size 2: the dark nodes are a dominating set, and the light nodes are another dominating set (the light nodes form a maximal independent set). See weak coloring for more information. Computational complexity Finding a domatic partition of size 1 is trivial: let . Finding a domatic partition of size 2 (or determining that it does not exist) is easy: check if there are isolated nodes, and if