Past exam of the graph theory course of the University of Oxford / 2011 / 2 / b / i / Solution

Take a proper colouring $c:V(G)\to \{1,\dots ,k\}$ and orient every edge from its lower-coloured endpoint towards its higher-coloured endpoint. Along a directed path the colour strictly increases, so such a graph path has at most $k$ vertices. A directed cycle is impossible for the same reason. Thus