Characteristic flows on signed graphs and short circuit covers

TL;DR

This paper extends Tutte's classical flow decomposition result to signed graphs, showing that any integer flow can be expressed as a sum of characteristic flows of signed circuits, with applications to flow coverings.

Contribution

It generalizes flow decomposition to signed graphs using signed circuits, including balanced and unbalanced types, and provides bounds on circuit coverings for graphs with nowhere-zero flows.

Findings

Flow decomposition extends to signed graphs with signed circuits.

Signed circuit coverings have length bounds related to flow parameters.

Application to flow coverings in signed graphs with nowhere-zero flows.

Abstract

We generalise to signed graphs a classical result of Tutte [Canad. J. Math. 8 (1956), 13--28] stating that every integer flow can be expressed as a sum of characteristic flows of circuits. In our generalisation, the r\^ole of circuits is taken over by signed circuits of a signed graph which occur in two types -- either balanced circuits or pairs of disjoint unbalanced circuits connected with a path intersecting them only at its ends. As an application of this result we show that a signed graph $G$ admitting a nowhere-zero $k$-flow has a covering with signed circuits of total length at most $2(k-1)|E(G)|$.