Embedding spanning subgraphs of small bandwidth

In this paper we prove the following conjecture by Bollobás and Komlós: For every γ > 0 and positive integers r and Δ, there exists β > 0 with the following property. If G is a sufficiently large graph with n vertices and minimum degree at least ( r − 1 r + γ ) n and H is an r-chromatic graph with n vertices, bandwidth at most βn and maximum degree at most Δ, then G contains a copy of H.