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.

Conjecture   Let be a -connected cubic graph and let be a -regular subgraph such that is connected. Then has a cycle double cover which contains (i.e all cycles of ).

Conjecture   (3-Decomposition Conjecture) Every connected cubic graph has a decomposition into a spanning tree, a family of cycles and a matching.

Conjecture   Let be a circuit in a bridgeless cubic graph . Then there is a five cycle double cover of such that is a subgraph of one of these five cycles.