2025/10/09 by Chao Wen, Qiang Sun, Wen, Chao +3
Computer Science · Mathematics · #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Stochastic processes and statistical mechanics #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2510.08192
openalex publication_date 2025/10/09 · openalex created_date 2025/10/18 · openalex updated_date 2026/07/28
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.