Degree-sensitive density bound

Establish 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 proposal by incorporating the number n3n_3 of degree-3 vertices. The paper explains that the conjecture was suggested because the known extremal examples for the simpler density bound contain many degree-3 vertices. If true, the conjecture would imply the 3-flow conjecture.

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 Conjecture (Li et al.), Section 1, Subsection 1.1

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 labeled “conj:li,” Section 1, subsection “Background and context”