Clustering powers of sparse graphs

03/07/2020

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by   Jaroslav Nešetřil, et al.

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We prove that if G is a sparse graph — it belongs to a fixed class of bounded expansion C— and d∈N is fixed, then the dth 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.