%0 Journal Article
%T On the total domatic number of regular graphs
%J Transactions on Combinatorics
%I University of Isfahan
%Z 2251-8657
%A Aram, H.
%A Sheikholeslami, S. M.
%A Volkmann, L.
%D 2012
%\ 03/01/2012
%V 1
%N 1
%P 45-51
%! On the total domatic number of regular graphs
%K total dominating set
%K total domination number
%K total domatic number
%K Regular graph
%R 10.22108/toc.2012.760
%X A 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 total domatic number 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 $r/(3\ln(r))$.
%U https://toc.ui.ac.ir/article_760_362cc9c41ad1def424bd149103450c49.pdf