The Codomatic Number of a Cubic Graph

The maximum cardinality of a partition of the vertex set of a graph \(G\) into dominating sets is the domatic number of \(G\), denoted \(d(G)\). The codomatic number of \(G\) is the domatic number of its complement, written \({d}(\overline{G})\). We show that the codomatic number for any cubic graph \(G\) of order \(n\) is \(n/2\), unless \(G \in \{K_4, G_1\}\) where \(G_1\) is obtained from \(K_{2,3} \cup K_3\) by adding the edges of a 1-factor between \(K_3\) and the larger partite set of \(K_{2,3}\).