Nowhere-zero flows on signed supereulerian graphs

TL;DR

This work advances Bouchet's conjecture by proving that every flow-admissible signed supereulerian graph with a spanning even Eulerian subgraph admits a nowhere-zero $6$-flow, and shows this bound is tight via a balanced Hamiltonian example with no $5$-flow. It introduces a sign-preserving reduction from 4-NZF-admissible graphs to 3-edge-colorable cubic graphs, establishing equivalences between admission of $k$-NZF across these classes and enabling a focused analysis on cubic and Hamiltonian structures. The authors apply the main result to signed abelian Cayley graphs, proving they all admit a $6$-flow (and identifying optimality in this setting), and they fully characterize the flow number $\\\Phi$ for odd-order abelian Cayley graphs in terms of the number of negative edges and the Cayley set size. These findings connect structural properties (spanning Eulerian subgraphs, Hamiltonian circuits) with flow extendability and provide precise flow-number criteria for a broad family of signed graphs.

Abstract

In 1983, Bouchet conjectured that every flow-admissible signed graph admits a nowhere-zero 6-flow. We verify this conjecture for the class of flow-admissible signed graphs possessing a spanning even Eulerian subgraph, which includes as a special case all signed graphs with a balanced Hamiltonian circuit. Furthermore, we show that this result is sharp by citing a known infinite family of signed graphs with a balanced Hamiltonian circuit that do not admit a nowhere-zero 5-flow. Our proof relies on a construction that transforms signed graphs whose underlying graph admits a nowhere-zero 4-flow into a signed 3-edge-colorable cubic graph. This transformation has the crucial property of establishing a sign-preserving bijection between the bichromatic cycles of the resulting signed cubic graph and certain Eulerian subgraphs of the original signed graph. As an application of our main result, we also show that Bouchet's conjecture holds for all signed abelian Cayley graphs.

Figures (13)

• Figure 1.1: $(G_{3},\sigma_{3})$.
• Figure 1.2: A flow-admissible signed abelian Cayley graph without any nowhere-zero $5$-flow.
• Figure 4.1: A $3$-NZF $f_{2}$ on $(H\cup \{e_{1},e_{2}\},\sigma)$.
• Figure 4.2: A $4$-NZF $f_{1}$ on $(G_{1},\sigma)$.
• Figure 5.1: A extending of $x_{i}x_{i+1}$ and $y_{i}y_{i+1}$, where $m$ is even.

Theorems & Definitions (47)

• Theorem 1.1: LMSZ25
• Definition 2.1
• Proposition 2.2
• Lemma 2.3
• Corollary 2.4
• Theorem 3.1
• Lemma 3.2
• proof
• Remark 3.3
• Theorem 3.4