The domatic number of regular and almost regular graphs

The domatic number of a graph G, denoted dom(G), is the maximum possible cardinality of a family of disjoint sets of vertices of G, each set being a dominating set of G. It is well known that every graph without isolated vertices has dom(G) ≥ 2. For every k, it is known that there are graphs with minimum degree at least k and with dom(G) = 2. In this paper we prove that this is not the case if G is k-regular or almost k-regular (by “almost” we mean that the minimum degree is k and the maximum degree is at most Ck for some fixed real number C ≥ 1). In this case we prove that dom(G) ≥ (1 + ok(1))k/(2 ln k). We also prove that the order of magnitude k/ lnk cannot be improved. One cannot replace the constant 2 with a constant smaller than 1. The proof uses the so called semi-random method which means that combinatorial objects are generated via repeated applications of the probabilistic method; in our case iterative applications of the Lovász Local Lemma.