SIAM Journal on Discrete Mathematics, Volume 34, Issue 4, Page 2166-2182, January 2020.
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.

中文翻译：

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没有长杠铃的签名图上的流

SIAM离散数学杂志，第34卷，第4期，第2166-2182页，2020年1月。
在Tutte的无符号图流理论中，许多基本属性没有与之对应的符号图。但是，从流的角度来看，没有长杠铃的带符号图在许多方面都表现得像无符号图。在本文中，我们研究了Tutte流动理论中的某些基本属性对于该签名图族是否仍然有效。具体来说，让$（G，\ sigma）$为允许流动的有符号曲线，而没有长的杠铃。我们证明了它接受无处零的6流，并且当且仅当它为每个整数$ k \ geq 3接受无处零的整数$ k $流时，它才接受无处零模$ k $流。 $和$ k \ not = 4 $。我们还表明，$（G，\ sigma）$的每个无零零正整数$ k $-流都可以表示为2个流的总和。对于一般图形，我们证明，每个无处不在的零$ \ frac {p} {q} $流都可以通过这种方式进行规范化，即每个流值都是$ \ frac {1} {2q} $的倍数。因此，对于没有长杠铃的可允许流量的有符号图，我们证明了整数流数与圆形流数的上限相等。