The Domatic Number of Regular Graphs

Abstract

The domatic number of a graph \(G\) is the maximum number of dominating sets into which the vertex set of \(G\) can be partitioned.

We show that the domatic number of a random \(r\)-regular graph is almost surely at most \(r\), and that for \(3\)-regular random graphs, the domatic number is almost surely equal to \(3\).

We also give a lower bound on the domatic number of a graph in terms of order, minimum degree, and maximum degree. As a corollary, we obtain the result that the domatic number of an \(r\)-regular graph is at least \((r+1)/(3ln(r+1))\).