The maximum number of disjoint dominating sets in a domatic partition of a graph 
is called its domatic number 
(Cockayne and Hedetniemi 1975, 1977).
The domatic number should not be confused with the domination number, which is the size of the smallest individual dominating set.
Let 
be the minimum vertex degree of a graph 
, then
 |
The domatic number of a graph with one or more isolated vertices is therefore 1.
Furthermore, if the domination number 
of a graph 
is known, than
 |
where 
denotes the vertex count of 
and 
is the floor function.
Finding the domatic number of a graph is computationally hard.
Given a complete set of minimal dominating sets, the domatic number of a graph 
can be found as the independence
number of the graph in which vertices are minimal dominating sets of 
and edges exist between pairs sets having nonempty intersection.
