# 3-Flows with Large Support

## TL;DR

This paper proves that every 3-edge-connected graph admits a 3-flow covering at least 5/6 of its edges, and shows this bound is tight with specific graph families.

## Contribution

It establishes the optimal lower bound of 5/6 for the support size of 3-flows in 3-edge-connected graphs, improving understanding of flow support sizes.

## Key findings

- Every 3-edge-connected graph has a 3-flow with support at least 5/6 of edges.
- The bound of 5/6 is proven to be tight using the graph K_4.
- An infinite family of graphs attains this bound, confirming its optimality.

## Abstract

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