Clustering Powers of Sparse Graphs

  • Jaroslav Nešetřil
  • Patrice Ossona de Mendez
  • Michał Pilipczuk
  • Xuding Zhu


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.

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