Domination in Bipartite Graphs and in Their Complements

The domatic numbers of a graph G and of its complement $bar G$ were studied by J. E. Dunbar, T. W. Haynes and M. A. Henning. They suggested four open problems. We will solve the following ones: Characterize bipartite graphs G having $dleft( G right) = dleft( {bar G} right)$ Further, we will present a partial solution to the problem: Is it true that if G is a graph satisfying $dleft( G right) = dleft( {bar G} right)$ then $gamma left( G right) = gamma left( {bar G} right)$ ? Finally, we prove an existence theorem concerning the total domatic number of a graph and of its complement