Nowhere-Zero Flows on Signed Wheels and Signed Fans.

A signed graph (G,σ)<inline-graphic></inline-graphic> is flow-admissible if there exists an orientation τ<inline-graphic></inline-graphic> and a positive integer k such that (G,σ)<inline-graphic></inline-graphic> admits a nowhere-zero k-flow. Bouchet (J Combin Theory Ser B 34:279-292, <xref>1983</xref>) 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. [ABSTRACT FROM AUTHOR]