Domination in Bipartite Graphs and in Their Complements

Abstract

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 $d\left( G \right) = d\left( {\bar G} \right)$

Further, we will present a partial solution to the problem:

Is it true that if G is a graph satisfying $d\left( G \right) = d\left( {\bar G} \right)$ 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