# Flows on signed graphs without long barbells

Y. Lu, R. Luo, M. Schubert, E. Steffen, C.-Q. Zhang, SIAM J. Discrete Math 34 (4) (2020) 2166–2187.

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Abstract
Many basic properties in Tutte's flow theory for unsigned graphs do not have their counterparts for signed graphs. However, signed graphs without long barbells in many ways behave like unsigned graphs from the point view of flows. In this paper, we study whether some basic properties in Tutte's flow theory remain valid for this family of signed graphs. Specifically let $(G,\sigma)$ be a flow-admissible signed graph without long barbells. We show that it admits a nowhere-zero $6$-flow and that it admits a nowhere-zero modulo $k$-flow if and only if it admits a nowhere-zero integer $k$-flow for each integer $k\geq 3$ and $k \not = 4$. We also show that each nowhere-zero positive integer $k$-flow of $(G,\sigma)$ can be expressed as the sum of some $2$-flows. For general graphs, we show that every nowhere-zero $\frac{p}{q}$-flow can be normalized in such a way, that each flow value is a multiple of $\frac{1}{2q}$. As a consequence we prove the equality of the integer flow number and the ceiling of the circular flow number for flow-admissible signed graphs without long barbells.
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Journal Title
SIAM J. Discrete Math
Volume
34 (4)
Page
2166-2187
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### Cite this

Lu Y, Luo R, Schubert M, Steffen E, Zhang C-Q. Flows on signed graphs without long barbells. SIAM J Discrete Math. 2020;34 (4):2166-2187. doi:10.1137/18M1222818
Lu, Y., Luo, R., Schubert, M., Steffen, E., & Zhang, C.-Q. (2020). Flows on signed graphs without long barbells. SIAM J. Discrete Math, 34 (4), 2166–2187. https://doi.org/10.1137/18M1222818
@article{Lu_Luo_Schubert_Steffen_Zhang_2020, title={Flows on signed graphs without long barbells}, volume={34 (4)}, DOI={10.1137/18M1222818}, journal={SIAM J. Discrete Math}, author={Lu, You and Luo, Rong and Schubert, Michael and Steffen, Eckhard and Zhang, Cun-Quan}, year={2020}, pages={2166–2187} }
Lu, You, Rong Luo, Michael Schubert, Eckhard Steffen, and Cun-Quan Zhang. “Flows on Signed Graphs without Long Barbells.” SIAM J. Discrete Math 34 (4) (2020): 2166–87. https://doi.org/10.1137/18M1222818.
Y. Lu, R. Luo, M. Schubert, E. Steffen, and C.-Q. Zhang, “Flows on signed graphs without long barbells,” SIAM J. Discrete Math, vol. 34 (4), pp. 2166–2187, 2020.
Lu, You, et al. “Flows on Signed Graphs without Long Barbells.” SIAM J. Discrete Math, vol. 34 (4), 2020, pp. 2166–87, doi:10.1137/18M1222818.

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