Question

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

• A. 8
• B. 4
• C. 3
• D. 2

Hint:

A cube's corners and edges form a bipartite graph (no odd cycles), and any bipartite graph needs exactly 2 colors.

Answer 1

The Correct Option is D

Step 1: Understand the graph.
The 8 corners of the cube, joined by the 12 edges of the cube, form a graph where we need the smallest number of colors so that no two corners joined directly by an edge share a color. This is the chromatic number of the cube graph.

Step 2: Recall the key idea for coloring by two colors.
A graph can be colored with only 2 colors exactly when it is bipartite, meaning its corners can be split into two groups such that every edge runs between the two groups, never within the same group. A graph is bipartite exactly when it contains no cycle of odd length.

Step 3: Split the cube's corners into two groups.
Label each corner of the cube with 3D coordinates of 0s and 1s, for example (0,0,0), (1,0,0), (0,1,0), and so on for all 8 corners, where each edge of the cube connects two corners that differ in exactly one coordinate.
Group 1: corners whose coordinates add up to an even number, such as (0,0,0), (1,1,0), (1,0,1), (0,1,1).
Group 2: corners whose coordinates add up to an odd number, such as (1,0,0), (0,1,0), (0,0,1), (1,1,1).
Every edge changes exactly one coordinate, so it always changes the sum from even to odd or odd to even, meaning every edge always goes from Group 1 to Group 2, never within the same group.

Step 4: Color the two groups.
Give every corner in Group 1 one color, say Red, and every corner in Group 2 a second color, say Blue. Since no edge connects two corners of the same group, no edge connects two corners of the same color. This coloring uses only 2 colors and is valid.

Step 5: Confirm that fewer than 2 colors is not possible.
Since the cube has at least one edge, its two endpoints must get different colors, so at least 2 colors are always needed. Combined with Step 4, exactly 2 colors are both necessary and sufficient.

Step 6: Rule out the other options.
8 and 4 use far more colors than needed, since the cube graph has cycles of length 4 (its square faces), not longer odd cycles that would force more colors. 3 colors would also work, but is not the minimum, since Step 4 already achieved a valid coloring with just 2.

Final Answer:
The minimum number of colors needed is 2. \[ \boxed{2} \]