Density bound for connected-flow-critical graphs

Establish that every connected-flow-critical graph on at least seven vertices satisfies $|E(G)|\le 3|V(G)|-8$.

Background

Connected-flow-critical graphs are defined by the failure to admit a nowhere-zero 3-flow while every nontrivial contraction whose parts induce connected subgraphs does admit one. Li et al. conjectured an upper bound of $3|V(G)|-8$ on the number of edges for graphs with at least seven vertices. The paper states explicitly that this conjecture remains unresolved.

References

Aside from the bounds given in , recent progress on lower bounds in and some progress when the genus of the graph is bounded in , Conjecture \ref{conj:densitycrit1} is still wide open.

— Flow-critical graphs  (2502.01451 - Árnadóttir et al., 3 Feb 2025) in Conjecture (conj:densitycrit1), Section 1, Subsection 1.1

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, Introduction