Generalizing well-known results of Erd ̋os and Lov ́asz, we show that every graph G contains a spanning k-partite subgraph H with λ(H) ≥ ⌈ k−1 k λ(G)⌉, where λ(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. 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.