We prove that if $G$ is a sparse graph --- it belongs to a fixed class of bounded expansion $\mathcal{C}$ --- and $d\in \mathbb{N}$ is fixed, then the $d$th power of $G$ can be partitioned into cliques so that contracting each of these clique to a single vertex again yields a sparse graph. This result has several graph-theoretic and algorithmic consequences for powers of sparse graphs, including bounds on their subchromatic number and efficient approximation algorithms for the chromatic number and the clique number.

中文翻译：

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稀疏图的聚类能力

我们证明如果$G$是一个稀疏图---它属于一个固定的有界展开$\mathcal{C}$---并且$d\in\mathbb{N}$是固定的，那么$可以将 $G$ 的第 d$ 次幂划分为群，以便将这些群中的每一个再次收缩为单个顶点，从而产生一个稀疏图。这个结果对稀疏图的幂有几个图论和算法的结果，包括它们的亚色数的界限和色数和团数的有效近似算法。