Remarks on nowhere-zero flows in signed cubic graphs

Abstract

We characterise signed cubic graphs that admit a nowhere-zero 3-flow, a nowhere-zero 4-flow, and nowhere-zero flows with values in abelian groups of order 3 and 4. Most of our characterisations feature the concept of an antibalanced signature, one that is switching-equivalent to the all-negative signature. In particular, we prove that a signed cubic graph has a nowhere-zero 3-flow if and only if it has a perfect matching and is antibalanced. Our results suggest several interesting problems for further investigation.