Nowhere-zero flow reconfiguration

TL;DR

This paper explores the reconfiguration of nowhere-zero flows in graphs, revealing both positive and negative results about their connectivity under various conditions and group structures.

Contribution

It introduces the concept of nowhere-zero flow reconfiguration, disproves a natural conjecture, and establishes new connectivity results for flows over different groups.

Findings

Not all 5-flows are connected in the reconfiguration sense.

All nowhere-zero $ ext{Z}_2^8$-flows are connected in 2-edge-connected graphs.

Connectivity depends on the group structure, not just the existence of flows.

Abstract

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 study this problem in the setting of integer flows and group flows, and prove a number of positive and negative results. * The natural reconfiguration variant of Tutte's 5-flow conjecture, stating that any two nowhere-zero 5-flows in any 2-edge-connected graph are connected, is false in the group and integer cases. * All nowhere-zero $\mathbb{Z}_2^8$-flows of every 2-edge-connected graph are connected and for every sufficiently large abelian group $A$, all nowhere-zero $A$-flows of every 2-edge-connected graph are connected. * The group structure affects the answer, contrary to the existence problem for nowhere-zero flows. * We 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. * For every 2-edge-connected graph $G$, there is an integer $k$ such that all nowhere-zero $k$-flows of $G$ are connected.