University of IsfahanTransactions on Combinatorics2251-86571120120301On the total domatic number of regular graphs455176010.22108/toc.2012.760ENH.AramAzarbaijan University of Tarbiat MoallemS. M.SheikholeslamiAzarbaijan University of Tarbiat Moallem0000-0003-2298-4744L.VolkmannRWTH-Aachen UniversityJournal Article20120112A set $S$ of vertices of a graph $G=(V,E)$ without isolated vertex is a total dominating set if every vertex of $V(G)$ is adjacent to some vertex in $S$. The <em>total domatic number</em> of a graph $G$ is the maximum number of total dominating sets into which the vertex set of $G$ can be partitioned. We show that the total domatic number of a random $r$-regular graph is almost surely at most $r-1$, and that for 3-regular random graphs, the total domatic number is almost surely equal to 2. We also give a lower bound on the total domatic number of a graph in terms of order, minimum degree and maximum degree. As a corollary, we obtain the result that the total domatic number of an $r$-regular graph is at least $r/(3\ln(r))$.https://toc.ui.ac.ir/article_760_362cc9c41ad1def424bd149103450c49.pdf