Conjecture (Berge and Fulkerson): Every 2-connected cubic graph has a collection of six perfect matchings that together cover every edge exactly twice. This conjecture is attributed to Berge in [2].

It is easy to see that an embedding is polyhedral if and only if the graph is 3-connected and the face-width is at least 3. The following conjecture is due to Grunbaum [1]. Conjecture (Grunbaum, 1969) ...