Forcing spanning subgraphs via Ore type conditions

We determine an Ore type condition that allows the embedding of 3-colourable bounded degree graphs of sublinear bandwidth: For all Δ,γ>0 there are β,n0>0 such that for all n⩾n0 the following holds. Let G=(V,E) and H be n-vertex graphs such that H is 3-colourable, has maximum degree Δ(H)⩽Δ and bandwidth bw(H)⩽βn, and G satisfies deg(u)+deg(v)⩾(4/3+γ)n for all uv∉E. Then G contains a copy of H. This improves on the Bollobás-Komlós conjecture for 3-chromatic graphs proven by Böttcher, Schacht, and Taraz [J. Combin. Theory, Ser. B, 98(4), 752–777, 2008] and applies a result of Kierstaed and Kostochka [J. Comb. Theory, Ser. B, 98(1), 226–234, 2008] about the existence of spanning triangle factors under Ore type conditions.