论文标题
$ 3 $ - 边缘连接性的集体连接性
Group Connectivity under $3$-Edge-Connectivity
论文作者
论文摘要
令$ s，t $为两个有限的Abelian组，$ | s | = | t | $。 Tutte的基本定理表明，当且仅当它接收到Noheyhere-Zero Zero $ t $ Flow时，图形允许一个零零$ s $ flow。 Jaeger，Linial，Payan和Tarsi于1992年引入了群体连接性作为流程理论的扩展，他们询问这种关系是否适合组连通性类比。 Hušek，Mohelníková和Šámal在2017年对其进行了负面答复。在本文中，我们将其结果扩展到$ 3 $与边缘连接的图（包括立方图和一般图），该图可以回答Hušek，Mohelníková和Šámal提出的开放问题（2017年）和Lai，Li，Li，Shao和Zhan（2011）。结合一些先前的结果，这表征了$ 3 $ - 边缘连接性的所有组连通性的所有等效性，表明每3 $ - 边缘连接$ s $ s $连接的图是$ t $ - 连接的，并且仅在$ \ \ {s，s，t \ \} \} \ neq \ neq \ neq \ neq \ neq \ {z \ mathbb {z} _ {z} $} $中
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 $\{S,T\}\neq \{\mathbb{Z}_4,\mathbb{Z}_2^2\}$.
