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We introduce classes of graphs with bounded expansion as a generalization of both proper minor closed classes and degree bounded classes. Such classes are based on a new invariant, the greatest reduced average density (grad) of G with rank r, ∇r(G). For these classes we prove the existence of several partition results such as the existence of low tree-width and low treedepth colorings. This generalizes and simplifies several earlier results (obtained for minor closed classes).

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No abstract