Show that a graph is bipartite if and only if all of its cycles are of even length.

Show that a bridgeless plane graph is bipartite if and only if all of its faces are of even length.

Let $G$ be an Eulerian plane graph. Show that the faces of $G$ can be coloured with two colours so that no two contiguous faces have the same colour. Deduce that it is possible to assign a direction to each edge of $G$ in such a way that the edges around each face form a directed cycle.