A $k$-subcolouring of a graph $G$ is a function $f:V(G) \to \{0,\ldots,k-1\}$ such that the set of vertices coloured $i$ induce a disjoint union of cliques. The subchromatic number, $\chi_{\textrm{sub}}(G)$, is the minimum $k$ such that $G$ admits a $k$-subcolouring. Ne\v{s}et\v{r}il, Ossona de Mendez, Pilipczuk, and Zhu (2020), recently raised the problem of finding tight upper bounds for $\chi_{\textrm{sub}}(G^2)$ when $G$ is planar. We show that $\chi_{\textrm{sub}}(G^2)\le 43$ when $G$ is planar, improving their bound of 135. We give even better bounds when the planar graph $G$ has larger girth. Moreover, we show that $\chi_{\textrm{sub}}(G^{3})\le 95$, improving the previous bound of 364. For these we adapt some recent techniques of Almulhim and Kierstead (2022), while also extending the decompositions of triangulated planar graphs of Van den Heuvel, Ossona de Mendez, Quiroz, Rabinovich and Siebertz (2017), to planar graphs of arbitrary girth. Note that these decompositions are the precursors of the graph product structure theorem of planar graphs. We give improved bounds for $\chi_{\textrm{sub}}(G^p)$ for all $p$, whenever $G$ has bounded treewidth, bounded simple treewidth, bounded genus, or excludes a clique or biclique as a minor. For this we introduce a family of parameters which form a gradation between the strong and the weak colouring numbers. We give upper bounds for these parameters for graphs coming from such classes. Finally, we give a 2-approximation algorithm for the subchromatic number of graphs coming from any fixed class with bounded layered cliquewidth. In particular, this implies a 2-approximation algorithm for the subchromatic number of powers $G^p$ of graphs coming from any fixed class with bounded layered treewidth (such as the class of planar graphs). This algorithm works even if the power $p$ and the graph $G$ is unknown.

翻译：图 $G$ 的 $k$-亚染色是指函数 $f:V(G) \to \{0,\ldots,k-1\}$，使得每种颜色 $i$ 的顶点集诱导出团的不交并。亚色数 $\chi_{\textrm{sub}}(G)$ 是使得 $G$ 存在 $k$-亚染色的最小 $k$ 值。Nešetřil、Ossona de Mendez、Pilipczuk 和 Zhu（2020）最近提出了当 $G$ 为平面图时 $\chi_{\textrm{sub}}(G^2)$ 紧上界的问题。我们证明当 $G$ 为平面图时 $\chi_{\textrm{sub}}(G^2)\le 43$，改进了他们 135 的界。当平面图 $G$ 具有更大围长时，我们给出了更优的界。此外，我们证明 $\chi_{\textrm{sub}}(G^{3})\le 95$，将先前 364 的界进行了改进。为此，我们借鉴了 Almulhim 和 Kierstead（2022）的一些最新技术，同时将 Van den Heuvel、Ossona de Mendez、Quiroz、Rabinovich 和 Siebertz（2017）的三角化平面图分解推广到任意围长的平面图。注意这些分解是平面图乘积结构定理的前身。对于所有 $p$，当 $G$ 具有有界树宽、有界简单树宽、有界亏格，或禁止包含团或双团作为子式时，我们给出了 $\chi_{\textrm{sub}}(G^p)$ 的改进界。为此，我们引入了一族参数，这些参数在强染色数与弱染色数之间形成梯度。我们给出了来自这些类的图的上界。最后，针对来自任意有界分层团宽类中的图，我们给出了亚色数的 2-近似算法。特别地，这意味着对于来自任意有界分层树宽类（如平面图类）的图 $G$ 的幂 $G^p$，亚色数存在 2-近似算法。该算法在幂 $p$ 和图 $G$ 未知的情况下仍然有效。