Flows of 3-edge-colorable cubic signed graphs

TL;DR

This paper proves that all flow-admissible 3-edge-colorable cubic signed graphs have a nowhere-zero 10-flow, extending to planar graphs and Hamiltonian graphs with specific flow bounds, advancing the understanding of signed graph flows.

Contribution

It establishes new flow bounds for 3-edge-colorable cubic signed graphs and related classes, partially confirming Bouchet's conjecture.

Findings

Every flow-admissible 3-edge-colorable cubic signed graph admits a nowhere-zero 10-flow.

Every flow-admissible bridgeless planar signed graph admits a nowhere-zero 10-flow.

Every flow-admissible Hamiltonian signed graph admits a nowhere-zero 8-flow.

Abstract

Bouchet conjectured in 1983 that every flow-admissible signed graph admits a nowhere-zero 6-flow which is equivalent to the restriction to cubic signed graphs. In this paper, we proved that every flow-admissible $3$-edge-colorable cubic signed graph admits a nowhere-zero $10$-flow. This together with the 4-color theorem implies that every flow-admissible bridgeless planar signed graph admits a nowhere-zero $10$-flow. As a byproduct, we also show that every flow-admissible hamiltonian signed graph admits a nowhere-zero $8$-flow.