The existence of tree-connected $\{g,f\}$-factors in edge-connected graphs and tough graphs

In 1970 Lov{\'a}sz gave a necessary and sufficient condition for the existence of a factor $F$ in a graph $G$ such that for each vertex $v$, $g(v)\le d_F(v)\le f(v)$, where $g$ and $f$ are two integer-valued functions on $V(G)$ with $g\le f$. In this paper, we give a sufficient edge-connectivity condition for the existence of an $m$-tree-connected factor $H$ in a bipartite graph $G$ with bipartition $(X,Y)$ such that its complement is $m_0$-tree-connected and for each vertex $v$, $d_H(v)\in \{g(v),f(v)\}$, provided that for each vertex $v$, $g(v)+m_0\le \frac{1}{2}d_G(v)\le f(v)-m$ and $|f(v)-g(v)|\le k$, and there is $h(v)\in \{g(v),f(v)\}$ in which $\sum_{v\in X}h(v)=\sum_{v\in Y}h(v)$. Moreover, we generalize this result to general graphs. As an application, we give sufficient conditions for the existence of tree-connected $\{g,f\}$-factors in edge-connected graphs and tough graphs.