Solution 4.4.5.1.

The cube can be represented as a planar graph and colored with two colors as follows:

$Eight vertices forming a inner and outer square (with edges forming the sides of each square), with edges connecting each inner vertex to its closest outer vertex neighbor. The outer vertices are labeled R, B, R, B (starting at the top left and moving clockwise). The inner vertices are labeled B, R, B, R (starting at the top left and moving clockwise).$

Since it would be impossible to color the vertices with a single color, we see that the cube has chromatic number 2 (it is bipartite).