Flow-critical graphs

TL;DR

This paper extends a fundamental theorem on nowhere-zero 3-flows in highly connected graphs, providing new conditions and results that advance understanding of flow extendability and contribute to longstanding conjectures.

Contribution

It generalizes Lovász et al.'s theorem to vertices with arbitrary degree under new conditions and explores minimal graphs related to flow extension failures.

Findings

Extended Lovász theorem to vertices with arbitrary degree

Identified conditions for flow extension failure in minimal graphs

Made progress on a conjecture of Li et al.

Abstract

Lov\'{a}sz et al. proved that every $6$-edge-connected graph has a nowhere-zero $3$-flow. In fact, they proved a more technical statement which says that there exists a nowhere zero $3$-flow that extends the flow prescribed on the incident edges of a single vertex $z$ with bounded degree. We extend this theorem of Lov\'{a}sz et al. to allow $z$ to have arbitrary degree, but with the additional assumption that there is another vertex $x$ with large degree and no small cut separating $x$ and $z$. Using this theorem, we prove two results regarding the generation of minimal graphs with the property that prescribing the edges incident to a vertex with specific flow does not extend to a nowhere-zero $3$-flow. We use this to further strengthen the theorem of Lov\'{a}sz et al., as well as make progress on a conjecture of Li et al.