  We study the exact square chromatic number of subcubic planar graphs. An
exact square coloring of a graph G is a vertex-coloring in which any two
vertices at distance exactly 2 receive distinct colors. The smallest number of
colors used in such a coloring of G is its exact square chromatic number,
denoted $\chi^{\sharp 2}(G)$. This notion is related to other types of
distance-based colorings, as well as to injective coloring. Indeed, for
triangle-free graphs, exact square coloring and injective coloring coincide. We
prove tight bounds on special subclasses of planar graphs: subcubic bipartite
planar graphs and subcubic K 4-minor-free graphs have exact square chromatic
number at most 4. We then turn our attention to the class of fullerene graphs,
which are cubic planar graphs with face sizes 5 and 6. We characterize
fullerene graphs with exact square chromatic number 3. Furthermore, supporting
a conjecture of Chen, Hahn, Raspaud and Wang (that all subcubic planar graphs
are injectively 5-colorable) we prove that any induced subgraph of a fullerene
graph has exact square chromatic number at most 5. This is done by first
proving that a minimum counterexample has to be on at most 80 vertices and then
computationally verifying the claim for all such graphs.
