The maximum number of disjoint dominating sets in a domatic partition of a graph 
is called its domatic number 
.
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 points 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.
