Periodic coloring of infinite planar graphs

The de Bruijn–Erdős theorem states that the number of colors needed to color an infinite graph is the same as the maximum number needed for its finite subgraphs. So for any reasonable definition of an infinite planar graph, the 4-color theorem for finite planar graphs implies that every infinite planar graph is also 4-colorable. One way to construct infinite planar graphs is to make them periodic: start with any periodic tiling of the plane, decorate a single prototile by vertices and edges that may wrap from one tile to the next, and form an infinite graph from the copies of these decorations on all the tiles of the tiling. One possibility is to simply use one vertex in each tile, with edges that connect that vertex to its copies in each adjacent tile, in which case coloring the graph is the same as coloring the tiles of the tiling. For instance, here are two periodic colorings of the floret pentagonal tiling, one with the same translational symmetries as the whole tiling and six…