Bounded census sum as a function of tip degree

Determine a function \(f:\mathbb{N}\to\mathbb{N}\) such that every tame flow-critical canvas (G,z) satisfies \(\sum_{n\in C(G,z)}n\leq f(\deg(z))\).

Background

This conjecture asks whether the total degree recorded in the census of a tame flow-critical canvas can be bounded solely in terms of the tip degree. The paper’s generation theorem gives algorithmic control for bounded tip degree, but the authors state that their proof method does not establish this conjecture because one generation operation can increase degrees without sufficient control.

References

There exists a function $f: \mathbb{N} \rightarrow \mathbb{N}$ such that if $(G,z)$ is a tame flow-critical canvas, then $$\sum_{n \in C(G,z)} n \leq f(\deg(z)).$$

— Flow-critical graphs  (2502.01451 - Árnadóttir et al., 3 Feb 2025) in Conjecture labeled “conj-censusplus,” Section 1, subsection “Results”