I can think of a way that to prove a 3-connected 5-regular planar graph does not contain a 5-critical subgraph.
We can choose two non-adjacent vertices a,b and contract them into a single vertex. If a and b has common neighbours, the resultant graph will still be a planar graph G'. But how can I prove chromatic number of G' is less or equal than 4?
Furthermore, how to prove that a 5-regular planar graph has chromatic number <= 4?