无处零流重配置 (Nowhere-zero flow reconfiguration)

We initiate the study of nowhere-zero flow reconfiguration. The natural question is whether any two nowhere-zero $k$-flows of a given graph $G$ are connected by a sequence of nowhere-zero $k$-flows of $G$, such that any two consecutive flows in the sequence differ only on a cycle of $G$. We conjecture that any two nowhere-zero 5-flows in any 2-edge-connected graph are connected in this way. This can be seen as a reconfiguration variant of Tutte's 5-flow conjecture. We study this problem in the setting of integer flows and group flows, and show that the structure of groups affects the answer, contrary to the existence of nowhere-zero flows. We also highlight a duality with recoloring in planar graphs and deduce that any two nowhere-zero 7-flows in a planar graph are connected, among other results. Finally we show that for any graph $G$, there is an abelian group $A$ such that all nowhere-zero $A$-flows in $G$ are connected, which is a weak form of our original conjecture. We conclude with several problems and conjectures.

翻译：我们首次开展无处零流重配置的研究。核心问题是：给定图$G$的任意两个无处零$k$-流，是否可以通过一系列$G$的无处零$k$-流相互连接，使得序列中任意两个相邻流仅在图$G$的一个环上存在差异。我们猜想：在任何2-边连通图中，任意两个无处零5-流均可通过此方式连接。这可视为Tutte五流猜想的重配置变体。我们在整数流与群流的框架下研究该问题，并证明群的结构会影响该问题的答案，这与无处零流的存在性结论形成对比。我们还揭示了该问题与平面图重着色问题的对偶性，并据此推导出平面图中任意两个无处零7-流均可相互连接等结论。最后我们证明：对任意图$G$，总存在一个阿贝尔群$A$，使得$G$中所有无处零$A$-流均可相互连接，这构成了原始猜想的一种弱形式。文末提出了若干待解决的问题与猜想。