  Barnette identified two interesting classes of cubic polyhedral graphs for
which he conjectured the existence of a Hamiltonian cycle. Goodey proved the
conjecture for the intersection of the two classes. We examine these classes
from the point of view of distance-two colorings. A distance-two $r$-coloring
of a graph $G$ is an assignment of $r$ colors to the vertices of $G$ so that
any two vertices at distance at most two have different colors. Note that a
cubic graph needs at least four colors. The distance-two four-coloring problem
for cubic planar graphs is known to be NP-complete. We claim the problem
remains NP-complete for tri-connected bipartite cubic planar graphs, which we
call type-one Barnette graphs, since they are the first class identified by
Barnette. By contrast, we claim the problem is polynomial for cubic plane
graphs with face sizes $3, 4, 5,$ or $6$, which we call type-two Barnette
graphs, because of their relation to Barnette's second conjecture. We call
Goodey graphs those type-two Barnette graphs all of whose faces have size $4$
or $6$. We fully describe all Goodey graphs that admit a distance-two
four-coloring, and characterize the remaining type-two Barnette graphs that
admit a distance-two four-coloring according to their face size.
  For quartic plane graphs, the analogue of type-two Barnette graphs are graphs
with face sizes $3$ or $4$. For this class, the distance-two four-coloring
problem is also polynomial; in fact, we can again fully describe all colorable
instances -- there are exactly two such graphs.
