arXiv 2205.090122022-05math.CO

有界度的模因子

Modulo factors with bounded degrees

Morteza Hasanvand

备注：: This paper is an improved version of a removed part of the paper arXiv:1702.07039

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AI总结：

本文研究了二部图及一般图的有界度模因子问题，证明了在特定边连通性和度数条件下图存在满足模同余及有界度约束的因子，并推广至一般图及正可整除度数的二部因子情形。

摘要：

设 $G$ 为具有二部划分 $(X,Y)$ 的二部图，$k$ 为正整数，$f:V(G)\rightarrow \{-1,\ldots, k-2\}$ 为满足 $\sum_{v\in X}f(v) \stackrel{k}{\equiv}\sum_{v\in Y}f(v)$ 的映射。本文证明，若 $G$ 本质上是 $(3k-3)$-边连通的，且对每个顶点 $v$ 满足 $d_G(v)\ge 2k-1+f(v)$，则其存在因子 $H$，使得对每个顶点 $v$，$d_H(v)\stackrel{k}{\equiv} f(v)$，且 $$\lfloor\frac{d_G(v)}{2}\rfloor-(k-1)\le d_{H}(v)\le \lceil\frac{d_G(v)}{2}\rceil+k-1.$$ 接着，我们将该结果推广至一般图，并推导出高度边连通一般图 $G$ 存在因子 $H$（满足对每个顶点 $v$ 有 $d_H(v)\in \{f(v),f(v)+k\}$）的充分条件。最后，我们证明每个 $(4k-1)$-边连通且本质 $(6k-7)$-边连通的图均包含度数为正且可被 $k$ 整除的二部因子。

英文摘要：

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 $k$.

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