On the bandwidth conjecture for 3-colourable graphs

A conjecture by Bollob´as and Koml´os states that for every γ > 0 and integers r ≥ 2 andΔ, there exists β > 0 such that for sufficiently large n the following holds: If G is a graph on n vertices with minimum degree at least ((r−1)/r +γ)n and H is an r-chromatic graph on n vertices with bandwidth at most βn and maximum degree at most Δ, then G contains a copy of H. This conjecture generalises several results concerning sufficient degree conditions for the containment of spanning subgraphs. We prove the conjecture for the case r = 3. Our proof yields a polynomial time algorithm for embedding H into G if H is given together with a 3-colouring and vertex labelling respecting the bandwidth bound.