Past exam of the probability, statistics and options I course of the University of Oxford / 2009 / R / 1 / ii

Partition the vertices by the parity of the number of one-bits. Every edge reverses that parity, giving a two-colouring; at least two colours are needed because an edge exists. Therefore the vertex chromatic number is two.

Every vertex has degree three, so its incident edges require three distinct colours. Three suffice by colouring an edge according to which of the three coordinates changes; every vertex has exactly one edge of each coordinate type. Thus the edge chromatic number is three.

The three faces incident to any cube vertex are pairwise adjacent along edges, forcing three colours. Three suffice by giving each pair of opposite faces a common colour, using a different colour for each of the three pairs. Hence the face chromatic number is three.