论文标题
高度与树连接的互补模量因子有界度
Highly tree-connected complementary modulo factors with bounded degrees
论文作者
论文摘要
令$ g $为带有两部分$（x，y）$的两部分图，让$ k $为一个正整数，让$ f：v（g）\ rightarrow z_k $是用$ \ sum_ {v \ in x} f（in x} f（v）f（v）f（v）\ stackrel \ stackrel {k} {在本文中，我们表明，如果$ g $是$（200万+2M_0+4K-4）$ - 边缘连接和$ m+m_0> 0 $，则$ g $具有$ $ m $ -tree连接的因子$ h $ $ d_h（v）\ stackrel {k} {\ equiv} f（v）$和$$ \ lfloor \ frac {d_g（v）} {2} {2} \ rfloor-（k-rfloor-（k-1）-m_0 \ lceil \ frac {d_g（v）} {2} \ rceil+k-1+m。$$接下来，我们将此结果推广到一般图表，并为具有高度边缘连接的一般图形$ g $得出足够的程度条件，以使每个顶点$ v $ v $ v $ v $ d_h（v）\ n $ h $ \ {f（v），f（v）+k \} $。最后，我们表明，每$（4K-2）$ - 连接的图形都承认了一个双方连接的因子，该因子的学位可除以$ k $。
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 $k$.
