  Generalizing well-known results of Erd\H{o}s and Lov\'asz, we show that every
graph $G$ contains a spanning $k$-partite subgraph $H$ with $\lambda{}(H)\geq
\lceil{}\frac{k-1}{k}\lambda{}(G)\rceil$, where $\lambda{}(G)$ is the
edge-connectivity of $G$. In particular, together with a well-known result due
to Nash-Williams and Tutte, this implies that every $7$-edge-connected graphs
contains a spanning bipartite graph whose edge set decomposes into two
edge-disjoint spanning trees. We show that this is best possible as it does not
hold for infintely many $6$-edge-connected graphs.
  For directed graphs, it was shown in [6] that there is no $k$ such that every
$k$-arc-connected digraph has a spanning strong bipartite subdigraph. We prove
that every strong digraph has a spanning strong 3-partite subdigraph and that
every strong semicomplete digraph on at least 6 vertices contains a spanning
strong bipartite subdigraph. \jbj{We generalize this result to higher
connectivities by proving} that, for every positive integer $k$, every
$k$-arc-connected digraph contains a spanning $(2k+1$)-partite subdigraph which
is $k$-arc-connected and this is best possible.
  A conjecture in [18] implies that every digraph of minimum out-degree $2k-1$
contains a spanning $3$-partite subdigraph with minimum out-degree at least
$k$. We prove that the bound $2k-1$ would be best possible by providing an
infinite class of digraphs with minimum out-degree $2k-2$ which do not contain
any spanning $3$-partite subdigraph in which all out-degrees are at least $k$.
  We also prove that every digraph of minimum semi-degree at least $3r$
contains a spanning $6$-partite subdigraph in which every vertex has in- and
out-degree at least $r$.
