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 $k$-次着色是指函数 $f:V(G) \to \{0,\ldots,k-1\}$，使得颜色 $i$ 的顶点集导出若干个团的不交并。次色数 $\chi_{\textrm{sub}}(G)$ 是使 $G$ 存在 $k$-次着色的最小 $k$ 值。Nešetřil、Ossona de Mendez、Pilipczuk 和 Zhu（2020）近期提出了当 $G$ 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 $\chi_{\textrm{sub}}(G^2)\le 43$，将他们的 135 界改进至此。我们给出了当平面图 $G$ 具有更大围长时的更优界。此外，我们证明 $\chi_{\textrm{sub}}(G^{3})\le 95$，将先前的 364 界改进至此。为得到这些结果，我们借鉴了 Almulhim 和 Kierstead（2022）近年发展的技术，同时将 Van den Heuvel、Ossona de Mendez、Quiroz、Rabinovich 和 Siebertz（2017）关于三角化平面图的分解推广至任意围长的平面图。注意这些分解是平面图积结构定理的前驱。对于任意 $p$，当 $G$ 具有有界树宽、有界简单树宽、有界亏格，或不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。为此引入了介于强染色数与弱染色数之间的一族参数，并给出此类图类的上界。最后，针对任意具有有界分层团宽固定图类中的图，我们给出了次色数的 2-近似算法。特别地，这蕴含了对于任意具有有界分层树宽固定图类（如平面图类）中图幂 $G^p$ 的次色数存在 2-近似算法。该算法在幂 $p$ 与图 $G$ 未知时仍适用。