Flows on signed graphs without long barbells

TL;DR

This paper investigates flow properties in signed graphs without long barbells, demonstrating they share key flow characteristics with unsigned graphs, including existence of certain flows and relationships between flow types.

Contribution

It extends Tutte's flow theory to a specific class of signed graphs, establishing new equivalences and flow decomposition results that mirror properties known for unsigned graphs.

Findings

Existence of nowhere-zero 6-flows in these graphs

Equivalence of modulo and integer flows for k ≥ 3, k ≠ 4

Flow values can be expressed as sums of 2-flows

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.