Abstract
A signed graph \((G, \sigma )\) is flow-admissible if there exists an orientation \(\tau \) and a positive integer k such that \((G, \sigma )\) admits a nowhere-zero k-flow. Bouchet (J Combin Theory Ser B 34:279–292, 1983) conjectured that every flow-admissible signed graph has a nowhere-zero 6-flow. In this paper, we show that each flow-admissible signed wheel admits a nowhere-zero 4-flow if and only if G is not the specified graph. Moreover, there are infinitely many signed wheels which do not admit a nowhere-zero 3-flow. We also prove each flow-admissible signed fan admits a nowhere-zero 4-flow. Similarly, there are infinitely many signed fans which do not admit a nowhere-zero 3-flow.
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