The Domatic Numbers of Factors of Graphs

The maximum cardinality of a partition of the vertex set of a graph \(G\) into dominating sets is the domatic number of \(G\), denoted \(d(G)\). We consider Nordhaus-Gaddum type results involving the domatic number of a graph, where a Nordhaus-Gaddum type result is a (tight) lower or upper bound on the sum or product of a parameter of a graph and its complement. Thereafter we investigate the upper bounds on the sum and product of the domatic numbers \(d(G_1), d(G_2)\) and \(d(G_3)\) where \(G_1 \oplus G_2 \oplus G_3 = K_n\). We show that the upper bound on the sum is \(n+2\), while the maximum value of the product is \(\lceil \frac{n}{3} \rceil ^3\) for \(n > 57\).