Consider the cube shown below with its 8 corners labelled a, b, c, d, e, f, g, and h. The figure is representative. All corners are to be colored such that any two corners that are connected by an edge must be of different colors. The minimum number of colors required to achieve this is ________
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Check whether the cube's corner and edge graph contains any cycle of odd length; if every cycle is even in length, the graph can be colored with just 2 colors.
Step 1: Model the cube as a graph. Treat each of the 8 corners (a, b, c, d, e, f, g, h) as a vertex, and each edge of the cube as a graph edge connecting two vertices. Every corner of a cube touches exactly 3 edges, so every vertex in this graph has degree 3, and the graph has 12 edges in total, matching the 12 edges of a cube.
Step 2: Recall what proper coloring means. We need to assign colors to the vertices so that no two vertices joined by an edge share the same color, using as few colors as possible. This smallest number of colors is called the chromatic number of the graph.
Step 3: Check whether the cube graph is bipartite. A graph is bipartite if its vertices can be split into two groups such that every edge goes between the two groups and never stays inside one group. For a cube, label each corner by how many edge steps it is from a chosen starting corner. Corners an even number of steps away form one group, and corners an odd number of steps away form the other group. Every face of a cube is a 4 sided square, and every closed path you can trace along the edges of a cube has an even number of edges (4, 6, 8, and so on), never an odd number. A graph with no odd length cycles is always bipartite.
Step 4: Use bipartiteness to color the cube with 2 colors. Split the 8 corners into two sets of 4 based on the even or odd step rule from Step 3 (this is the same split as the two sets of 4 corners that form the two tetrahedra inscribed inside a cube). Color every vertex in the first set with Color 1 and every vertex in the second set with Color 2. Since every edge of the cube always connects one vertex from the first set to one vertex from the second set, no edge ever joins two vertices of the same color, so this 2-coloring is valid.
Step 5: Show that 1 color is not enough. If only 1 color were used, every pair of connected vertices, for example a and b which share an edge, would end up with the same color, violating the rule that connected corners must differ. So at least 2 colors are required.
Step 6: Combine the bounds. Step 5 shows at least 2 colors are necessary, and Step 4 shows 2 colors are sufficient to properly color all 8 corners. So the minimum number of colors required is exactly 2, meaning options (A) 8, (B) 4 and (C) 3 all use more colors than necessary.
