Integer Flows and Modulo Orientations of Signed Graphs

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 $p \geq 3$.