Modulo factors with bounded degrees

Abstract

Let $G$ be a bipartite graph with bipartition $(X,Y)$, let $k$ be a positive integer, and let $f:V(G)\to \{-1,\dots ,k-2\}$ 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 $k$.

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