论文标题
具有有限度的模型因子
Modulo factors with bounded degrees
论文作者
论文摘要
令$ g $为带有两部分$（x，y）$的两项图，让$ k $为正整数，让$ f：v（g）\ rightarrow \ rightarrow \ { - 1，\ ldots，k-2 \} $是$ \ sum_ {v \ in x} f（v）f（v）f（v）， \ stackrel {k} {\ equiv} \ sum_ {v \ in y} f（v）$。在本文中，我们表明，如果$ g $本质上是$（3K-3）$ - 边缘连接，并且对于每个顶点$ v $，$ d_g（v）\ ge 2k-1+f（v）$ $ \ lfloor \ frac {d_g（v）} {2} \ rfloor-（k-1）\ le d_ {h}（h}（v）\ le \ le \ lceil \ lceil \ frac {d_g（v）} $ g $具有因子$ h $，以便每个顶点$ v $，$ d_h（v）\ in \ {f（v），f（v）+k \} $。最后，我们表明，每$（4K-1）$ - 边缘连接基本上是$（6K-7）$ - 边缘连接的图，可以承认一个两分因子，其学位是正面的，可以由$ k $排除。
Let $G$ be a bipartite graph with bipartition $(X,Y)$, let $k$ be a positive integer, and let $f:V(G)\rightarrow \{-1,\ldots, k-2\}$ 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 essentially $(3k-3)$-edge-connected and for each vertex $v$, $d_G(v)\ge 2k-1+f(v)$, then it admits a factor $H$ such that for each vertex $v$, $d_H(v)\stackrel{k}{\equiv} f(v)$, and $$\lfloor\frac{d_G(v)}{2}\rfloor-(k-1)\le d_{H}(v)\le \lceil\frac{d_G(v)}{2}\rceil+k-1.$$ Next, we generalize this result to general graphs and derive sufficient conditions for a highly edge-connected general graph $G$ to have a factor $H$ such that for each vertex $v$, $d_H(v)\in \{f(v),f(v)+k\}$. Finally, 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 $k$.
