2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/76060We 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, grad r(G). For these classes we prove the existence of several partition results such as the existence of low tree-width and low tree-depth colorings. This generalizes and simplifies several earlier results (obtained for minor closed classes).Combinatorics05C15 05C75text