Nowhere-zero flow reconfiguration

TL;DR

The paper initiates the study of nowhere-zero flow reconfiguration, defining reconfiguration graphs for flows over abelian groups and integers and investigating when these graphs are connected. It reveals that the group structure of the flows crucially affects connectivity, with strong planar duality links to recoloring that transfer connectivity results from colorings to flows. Several broad results are established: (i) connectivity for large product groups on general and planar graphs via cycle-based moves and dual degeneracy arguments, (ii) a tight equivalence between $\mathcal{F}(G,\mathbb{Z}_2\times\mathbb{Z}_2)$ and Kempe-change connectivity in cubic planar graphs, and (iii) a general theorem that $\\mathcal{F}(G,A\times B)$ is connected with linear diameter for large $A,B$ on graphs where every edge lies in small cycles. The paper concludes with detailed observations, conjectures about composite flows, and directions for future work on boundaries, diameter, and component structure.

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 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.

Figures (8)

• Figure 1: Illustration of a $\mathbb{Z}_4$-flow in $K_4$ and the only cycle along which some flow value can be added.
• Figure 2: The reconfiguration graph $\mathcal{F}(G, 4)$
• Figure 3: A 3-edge-connected cubic graph $G$ with a nowhere-zero $\mathbb{Z}_5$-flow $f$ having a unique neighbor $g$ in $\mathcal{F}(G, \mathbb{Z}_5)$. The support of $g-f$ is highlighted in bold.
• Figure 4: The nowhere-zero $\mathbb{Z}_4$-flows of $G$ are in bijection with the nowhere-zero $\mathbb{Z}_4$-flows of $H$. The case where $v$ has outdegree 2 is symmetric.
• Figure 5: A plane (oriented) graph $G$ and its dual (oriented) graph $G^*$. Any proper coloring of $G^*$ induces a nowhere-zero flow in $G$.

Theorems & Definitions (48)

• Conjecture 1.1: The 5-flow reconfiguration conjecture
• Conjecture 1.2
• Conjecture 1.3
• Conjecture 1.4
• Theorem 2.1: Tutte190
• Theorem 2.2: Tutte190
• proof
• Theorem 2.5: BonsmaCereceda2007degeneracy2006
• Lemma 3.1
• proof