Häggkvist–Markström Kotzig spanning-minor conjecture

A cubic graph is a graph in which every vertex has degree three. A cubic graph $H$ is a spanning minor of a cubic graph $G$ if some subdivision of $H$ is a spanning subgraph of $G$. A Kotzig graph is a cubic graph with a 3-edge-coloring such that the union of any two color classes is a Hamilton circuit.

Häggkvist–Markström conjecture. Every $3$-connected cubic graph contains a Kotzig graph as a spanning minor.

The conjecture would provide a broad sufficient condition for the circuit double cover conjecture, because a cubic graph containing a Kotzig graph as a spanning minor has a $6$-even-subgraph double cover. The source also notes that $3$-edge-connectivity is insufficient and that cyclical $4$-edge-connectivity may be necessary.