# Signed graphs: from modulo flows to integer-valued flows

## TL;DR

This paper explores how to convert modulo flows into integer-valued flows in signed graphs, extending previous results and showing that certain modulo flows can always be extended to integer flows.

## Contribution

It generalizes earlier results by demonstrating that modulo
(2+1/p)-flows in signed graphs can be extended to integer-valued flows, filling a gap in the theory.

## Key findings

- Every modulo (2+1/p)-flow in signed graphs can be extended to an integer-valued flow.
- Generalizes previous results on flow conversions for signed graphs.
- Provides a framework for converting modulo flows to integer flows in more complex graph structures.

## Abstract

Converting modulo flows into integer-valued flows is one of the most critical steps in the study of integer flows. Tutte and Jaeger's pioneering work shows the equivalence of modulo flows and integer-valued flows for ordinary graphs. However, such equivalence does not hold any more for signed graphs. This motivates us to study how to convert modulo flows into integer-valued flows for signed graphs. In this paper, we generalize some early results by Xu and Zhang (Discrete Math.~299, 2005), Schubert and Steffen (European J. Combin.~48, 2015), and Zhu (J. Combin. Theory Ser. B~112, 2015), and show that, for signed graphs, every modulo $(2+\frac{1}{p})$-flow with $p \in {\mathbb Z}^+ \cup \{\infty\}$ can be converted/extended into an integer-valued flow.