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Let G be a graph with X ⊆ V (G) and let l be an intersecting supermodular subadditive integer-valued function on subsets of V (G). The graph G is said to be l-partition-connected, if for every partition P of, where eG(P ) denotes the number of edges of G joining different parts of P . Let λ ∈ [0, 1] be a real number and let η be a real function on X. In this paper, we show that if G is l-partition-connected and for all S ⊆ X,then G has an l-partition-connected spanning subgraph H such that for each vertex v ∈ X, dH (v) ≤ ⌈η(v) − λl(v)⌉, where eG(S) denotes the number of edges of G with both ends in S and Θ l (G \ S) denotes the maximum number of all A∈P l(A) − e G\S (P ) taken over all partitions P of V (G) \ S. Finally, we show that if H is an (l1 + · · · + lm)-partition-connected graph, then it can be decomposed into m edge-disjoint spanning subgraphs H1, . . . , Hm such that every graph Hi is li-partition-connected, where l1, l2, . . . , lm are m intersecting supermodular subadditive integer-valued functions on subsets of V (H).These results generalize several known results.

Let G be a bipartite graph with bipartition (X, Y ), let k be a positive integer, and let f :Next, we generalize this result to general graphs and derive a sufficient degree condition for a highly edge-connected general graph G to have a factor H such that for each vertex v, dHFinally, we show that every (4k − 1)-edge-connected essentially (6k − 7)-edge-connected graph admits a bipartite factor whose degrees are positive and divisible by k.

Let G be a 2k‐edge‐connected graph with k≥0 and let Lfalse(vfalse)⊆{k,…,dGfalse(vfalse)} for every v∈V(G). A spanning subgraph F of G is called an L‐factor, if dFfalse(vfalse)∈Lfalse(vfalse) for every v∈V(G). In this article, we show that if false|L(v)false|≥false⌈dG(v)2false⌉+1 for every v∈V(G), then G has a k‐edge‐connected L‐factor. We also show that if k≥1 and Lfalse(vfalse)={⌊dG(v)2⌋,…,false⌈dG(v)2false⌉+k} for every v∈V(G), then G has a k‐edge‐connected L‐factor.

Let G be a graph, let k be a positive integer, and let p : V (G) → Z k be a mapping withThis result reduces the required edge-connectivity of several results toward decomposing a graph into isomorphic copies of a fixed tree. Next, we conclude that if G is a (3k − 3)-edge-connected bipartite graph with the bipartition (A, B), then it has a factor H such that for each vertex v,. Finally, we investigate decomposition of highly edge-connected graphs into factors with bounded degrees with many edge-disjoint spanning trees and deduce that every 4-edge-connected graph G has a spanning Eulerian subgraph whose degrees are close to dG(v)/2. As a consequence, every 4-edge-connected 10-regular graph has a spanning Eulerian subgraph whose degrees lie in the set {4, 6}.