Spanning 3-colourable subgraphs of small bandwidth in dense graphs

A conjecture by Bollobás and Komlós states the following: For every γ > 0 and integers r ⩾ 2 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. onjecture generalises several results concerning sufficient degree conditions for the containment of spanning subgraphs. We prove the conjecture for the case r = 3 .