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 , n 0 > 0 such that for all n ⩾ n 0 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 ) ⩽ β n , and G satisfies deg ( u ) + deg ( v ) ⩾ ( 4 3 + γ ) n for all u v ∉ E . Then G contains a copy of H. mproves 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.