Group Connectivity under $3$-Edge-Connectivity

Let $S,T$ be two distinct finite Abelian groups with $|S|=|T|$. A fundamental theorem of Tutte shows that a graph admits a nowhere-zero $S$-flow if and only if it admits a nowhere-zero $T$-flow. Jaeger, Linial, Payan and Tarsi in 1992 introduced group connectivity as an extension of flow theory, and they asked whether such a relation holds for group connectivity analogy. It was negatively answered by Hušek, Mohelníková and Šámal in 2017 for graphs with edge-connectivity 2 for the groups $S=\mathbb{Z}_4$ and $T=\mathbb{Z}_2^2$. In this paper, we extend their results to $3$-edge-connected graphs (including both cubic and general graphs), which answers open problems proposed by Hušek, Mohelníková and Šámal(2017) and Lai, Li, Shao and Zhan(2011). Combining some previous results, this characterizes all the equivalence of group connectivity under $3$-edge-connectivity, showing that every $3$-edge-connected $S$-connected graph is $T$-connected if and only if $\{S,T\}\neq \{\mathbb{Z}_4,\mathbb{Z}_2^2\}$.

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