The natural orientation conjecture for finite graphs

A finite graph $\Gamma$ is called naturally oriented if $H^1(\Gamma)$ has a basis whose supports are isomorphic, in the orientation-preserving sense, to cycles whose edges all point in the same direction; these supports are called primitive cycles. Natural orientation conjecture. Every finite graph has a natural orientation. Natural orientations would provide cohomology bases represented by edge forms with only $0$ and $1$ entries, giving a canonical combinatorial structure relevant to the action of the automorphism group on graph cohomology and to quantum mechanics on graphs. The source provides no resolution of the conjecture.