Total domatic number of a graph. I (Q1265335)

The total domatic number \(d_t\) of an undirected, simple graph \(G\) is the maximum order of a partition of \(V(G)\) into classes such that each class \(C\) is a total dominating set in \(G\), i.e. each vertex of \(V(G)\) is adjacent to some vertex of \(C\). Using results of \textit{E. J. Cockayne}, \textit{R. M. Dawes} and \textit{S. T. Hedetniemi} [Networks 10, 211-219 (1980; Zbl 0447.05039)], the authors characterize all regular graphs \(G\) fulfilling \(d_t+\overline d_t= p-2\), where \(p\) denotes the order of \(G\) and \(\overline d_t\) the total domatic number of the complement \(\overline G\) of \(G\).