Census characterization with a degree-(deg(z)−3) vertex

Prove that if a non-trivial flow-critical tame canvas (G,z) has deg(z) ≥ 8 and contains a vertex of degree deg(z)−3, then its census is either {deg(z)−3,5} or {deg(z)−3,5,5,5}.

Background

The paper proves this census characterization under the additional assumption that a suitable tip preflow does not extend to a nowhere-zero flow. The conjecture removes that assumption and would, together with the paper’s theorem for degree deg(z)−2, sharply restrict the possible censuses when the tip has degree eight.

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 Conjecture labeled “conj-db3,” Section “Bounding the censuses: Proving Theorem \ref{thm-degbetter}”