Census classification with a degree-three gap

Characterize every non-trivial tame flow-critical canvas $(G,z)$ with $\deg(z)\geq 8$ and a vertex of degree $\deg(z)-3$, proving that its census is either $\{\deg(z)-3,5\}$ or $\{\deg(z)-3,5,5,5\}$.

Background

The paper proves a conditional version of this statement when there exists a tip preflow that does not extend to a nowhere-zero flow and that preflow is not uniform on the edges from the tip to the other non-tip vertices. The conjecture removes that additional preflow assumption. The authors explain that proving it would further classify possible censuses, particularly when the tip has degree 8.

References

We believe the assumption of the existence of the tip preflow $\psi$ can be dropped. Let $(G,z)$ be a non-trivial flow-critical tame canvas such that $\deg(z)\ge 8$. If $G$ has a vertex of degree $\deg(z)-3$, then $C(G,z)$ is either ${\deg(z)-3,5}$ or ${\deg(z)-3,5,5,5}$.

— Flow-critical graphs  (2502.01451 - Árnadóttir et al., 3 Feb 2025) in Section 7, Subsection 7.2 (Containing the censuses), Conjecture \ref{conj-db3}