Degree-sensitive density bound for connected-flow-critical graphs

Prove that every connected-flow-critical graph on at least seven vertices with $n_3$ vertices of degree 3 satisfies $|E(G)| < \frac{5|V(G)|}{2}+n_3$.

Background

This conjecture strengthens the preceding density conjecture by incorporating the number of degree-3 vertices. The paper explains that this stronger inequality would imply the 3-flow conjecture, making it a particularly consequential unresolved problem concerning the structure of connected-flow-critical graphs.

References

For any connected-flow-critical graph on at least seven vertices with $n_{3}$ vertices of degree $3$, we have $$|E(G)| < \frac{5|V(G)|}{2} +n_{3}.$$

— Flow-critical graphs  (2502.01451 - Árnadóttir et al., 3 Feb 2025) in Section 1, Subsection 1.1, Conjecture by Li et al.

Nevertheless, these examples have many vertices of degree 3, and hence the authors of suggest the following, which does imply the $3$-flow conjecture. For any connected-flow-critical graph on at least seven vertices with $n_{3}$ vertices of degree $3$, we have $|E(G)| < \frac{5|V(G)|}{2} +n_{3}$.

— Flow-critical graphs  (2502.01451 - Árnadóttir et al., 3 Feb 2025) in Conjecture, label conj:li, Section 1, subsection “Background and context”