Conjecture   Let be a cubic graph with a nz--flow, and let each edge satisfying does not have a nz--flow be colored black, and let each edge satisfying has a nz--flow be colored white. Then contains a white cycle but not a black cycle.

Used terminology: nz--flow is the abbrevation for nowhere-zero -flow, a cycle is a connected -regular graph, a cycle is called black (white) if each edge of the cycle is colored black (white).

Conjecture   Suppose with is a connected cubic graph admitting a -edge coloring. Then there is an edge such that the cubic graph homeomorphic to has a -edge coloring.