Pairs of Disjoint Dominating Sets and the Minimum Degree of Graphs

For a connected graph G of order n and minimum degree δ we prove the existence of two disjoint dominating sets D ₁ and D ₂ such that, if δ ≥ 2, then ${|D_1cup D_2|leq frac{6}{7}n}$ unless G = C ₄, and, if δ ≥ 5, then ${|D_1cup D_2|leq 2frac{1+ln(delta+1)}{delta+1}n}$ . While for the first estimate there are exactly six extremal graphs which are all of order 7, the second estimate is asymptotically best-possible.