Tree decompositions of graphs without large bipartite holes

A recent result of Condon, Kim, Kuhn, and Osthus implies that for anyr >=(12+o(1))n, ann-vertex almostr-regular graphGhas an approximate decomposition into any collections ofn-vertex bounded degree trees. In this paper, we prove that a similar result holds for an almost alpha n-regular graphGwith any alpha>0 and a collection of bounded degree trees on at most (1-o(1))nvertices ifGdoes not contain large bipartite holes. This result is sharp in the sense that it is necessary to exclude large bipartite holes and we cannot hope for an approximate decomposition inton-vertex trees. Moreover, this implies that for any alpha>0 and ann-vertex almost alpha n-regular graphG, with high probability, the randomly perturbed graphG?G(n,O(1n))has an approximate decomposition into all collections of bounded degree trees of size at most (1-o(1))nsimultaneously. This is the first result considering an approximate decomposition problem in the context of Ramsey-Turan theory and the randomly perturbed graph model.