SIAM Journal on Discrete Mathematics, Volume 35, Issue 1, Page 376-391, January 2021.
This paper studies the fundamental relations among integer flows, modulo orientations, integer-valued and real-valued circular flows, and monotonicity of flows in signed graphs. A (signed) graph is modulo-$(2p+1)$-orientable if it has an orientation such that the indegree is congruent to the outdegree modulo $2p+1$ at each vertex. An integer-valued $\frac{2p+1}{p}$-flow is a flow taking integer values in $\{\pm p, \pm (p+1)\}$. Extending a fundamental result of Jaeger to signed graphs, we show that a bridgeless signed graph is modulo-$(2p+1)$-orientable if and only if it admits an integer-valued $\frac{2p+1}{p}$-flow. It was conjectured by Raspaud and Zhu that, for any signed graph, the admission of a circular $r$-flow implies the admission of an integer-valued $\lceil r \rceil$-flow. Although this conjecture has been disproved in general, it is confirmed in this paper for bridgeless signed graphs if $r=\frac{2p+1}{p}$ and $p \geq 3$.

中文翻译：

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有符号图的整数流和模方向

SIAM 离散数学杂志，第 35 卷，第 1 期，第 376-391 页，2021 年 1 月。
本文研究了整数流、模方向、整数值和实值循环流以及有符号图中流的单调性之间的基本关系。一个（有符号）图是模$(2p+1)$-orientable，如果它的方向使得入度与每个顶点的出度模$2p+1$一致。整数值 $\frac{2p+1}{p}$-flow 是在 $\{\pm p, \pm (p+1)\}$ 中取整数值的流。将 Jaeger 的基本结果扩展到有符号图，我们证明无桥有符号图是模-$(2p+1)$-orientable 当且仅当它承认一个整数值 $\frac{2p+1}{p} $-流量。Raspaud 和 Zhu 推测，对于任何有符号图，循环 $r$-flow 的接受意味着整数值 $\lceil r \rceil$-flow 的接受。虽然这个猜想已被普遍推翻，