Let ${\rm Lct}(G)$ denote the set of all lengths of closed trails that exist in an even graph $G$. A sequence $(t_1,\dots ,t_p)$ of elements of ${\rm Lct}(G)$ adding up to $|E(G)|$ is $G$-realisable provided there is a sequence $(T_1,\dots ,T_p)$ of pairwise edge-disjoint closed trails in $G$ such that $T_i$ is of length $t_i$ for $i=1,\dots ,p$. The graph $G$ is arbitrarily decomposable into closed trails if all possible sequences are $G$-realisable. In the paper it is proved that if $a\ge 1$ is an odd integer and $M_{a,a}$ is a perfect matching in $K_{a,a}$, then the graph $K_{a,a}-M_{a,a}$ is arbitrarily decomposable into closed trails.
The domatic numbers of a graph $G$ and of its complement $\bar{G}$ were studied by J. E. Dunbar, T. W. Haynes and M. A. Henning. They suggested four open problems. We will solve the following ones: Characterize bipartite graphs $G$ having $d(G) = d(\bar{G})$. Further, we will present a partial solution to the problem: Is it true that if $G$ is a graph satisfying $d(G) = d(\bar{G})$, then $\gamma (G) = \gamma (\bar{G})$? Finally, we prove an existence theorem concerning the total domatic number of a graph and of its complement.