Highly tree-connected complementary modulo factors with bounded degrees

Abstract: Let $G$ be a bipartite graph with bipartition $(X,Y)$, let $k$ be a positive integer, and let $f:V(G)\rightarrow Z_k$ be a mapping with $\sum_{v\in X}f(v) \stackrel{k}{\equiv}\sum_{v\in Y}f(v)$. In this paper, we show that if $G$ is $(2m+2m_0+4k-4)$-edge-connected and $m+m_0>0$, then $G$ has an $m$-tree-connected factor $H$ such that its complement is $m_0$-tree-connected and for each vertex $v$, $d_H(v)\stackrel{k}{\equiv} f(v)$, and $$\lfloor\frac{d_G(v)}{2}\rfloor-(k-1)-m_0\le d_{H}(v)\le \lceil\frac{d_G(v)}{2}\rceil+k-1+m.$$ 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 connected factor $H$ such that for each vertex $v$, $d_H(v)\in {f(v),f(v)+k}$. Finally, we show that every $(4k-2)$-tree-connected graph admits a bipartite connected factor whose degrees are divisible by $k$.