3-Flows with Large Support

We prove that every 3-edge-connected graph $G$ has a 3-flow $ϕ$ with the property that $|\mathop{supp}(ϕ)| \ge \frac{5}{6} |E(G)|$. The graph $K_4$ demonstrates that this $\frac{5}{6}$ ratio is best possible; there is an infinite family where $\frac 56$ is tight.

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