Group Connectivity, Strongly Z_m-Connectivity, and Edge Disjoint Spanning Trees

Abstract

Let $\mathbb Z_m$ be the cyclic group of order $m \geq 3$. A graph $G$ is $\mathbb Z_m$-connected if $G$ has an orientation $D$ such that for any mapping $b: V(G) \mapsto \mathbb Z_m$ with $\sum_{v\in V(G)}b(v)=0$, there exists a mapping $f:E(G) \mapsto \mathbb Z_m -\{0\}$ satisfying $\sum_{e\in E_D^+(v)} f(e) - \sum_{e\in E_D^-(v)} f(e) = b(v)$ in $\mathbb Z_m$ for any $v \in V(G)$; and a graph $G$ is strongly $\mathbb Z_m$-connected if, for any mapping $\theta: V(G)\rightarrow \mathbb Z_m$ with $\sum_{v\in V(G)}\theta(v) = |E(G)|$ in $\mathbb Z_m$, there is an orientation $D$ such that $d_D^+(v)=\theta(v)$ in $\mathbb Z_m$ for each $v \in V(G)$. In this paper, we study the relation between $\mathbb Z_m$-connected graphs and strongly $\mathbb Z_m$-connected graphs and show that a graph $G$ is $\mathbb Z_m$-connected if and only if $(m-2)G$ is strongly $\mathbb Z_m$-connected, where $(m-2)G$ is the graph obtained from $G$ by replacing each edge in $G$ with $m-2$ parallel edges. We also show that if $G$ is...