Is it possible to have a planar graph with a chromatic number of $4$ such that all vertices have degree $4$?
Every time I try to make the degree condition to work on a graph, it loses its planarity.
Here is an example.
The graph is 4 regular and 4 colorable. However, it's not possible to color its vertices with 3 colors. The picture shows a partial coloring that cannot be extended to the whole graph.