Tutte's three nowhere-zero flow conjectures

Let $G$ be a graph. A nowhere-zero $\mathbb{Z}_k$-flow is a flow whose values lie in $\mathbb{Z}_k$ and are nonzero on every edge. The graph is $r$-edge-connected if every edge cut has size at least $r$.

Tutte's three flow conjectures. The following three statements all hold:

• Every $2$-edge-connected graph has a nowhere-zero $\mathbb{Z}_5$-flow.
• Every $2$-edge-connected graph with no Petersen minor has a nowhere-zero $\mathbb{Z}_4$-flow.
• Every $4$-edge-connected graph has a nowhere-zero $\mathbb{Z}_3$-flow.

Despite partial results, all three conjectures remain open.