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.

中文翻译：

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亚三次平面图的精确正方形着色 *

我们研究了亚三次平面图的精确平方色数。图 G 的精确正方形着色是一种顶点着色，其中距离恰好为 2 的任意两个顶点接收不同的颜色。在 G 的这种着色中使用的最小颜色数是其精确的平方色数，表示为 $\chi^{\sharp 2}(G)$。这个概念与其他类型的基于距离的着色以及单射着色有关。事实上，对于无三角形图，精确的正方形着色和单射着色是一致的。我们证明了平面图的特殊子类的紧边界：亚三次二部平面图和亚三次 K 4-minor-free 图的精确平方色数最多为 4。然后我们将注意力转向富勒烯图的类，它们是三次平面图面部尺寸为 5 和 6。