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 ∆, 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 ∆(H) ≤ ∆ and bandwidth bw(H) ≤ βn, and G satisfies deg(u) + deg(v) ≥ ( 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.