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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For an ideal gas, which of the following thermodynamic quantities depends only on temperature?

Answer 1

Another way to find the minimum number of colors is to look directly at the cycles (closed loops of edges) in the cube graph, since the length and parity of those cycles decides the chromatic number for a graph like this one.

Look at the shortest loops: every face of the cube is a square, so walking around any one face traces a closed loop of exactly 4 edges. A loop of even length, like 4, never forces more than 2 colors on its own, since you can alternate two colors all the way around and land back on the start with no clash.
Check for odd loops: a graph needs 3 or more colors only if it contains at least one cycle with an odd number of edges (a triangle, a 5-loop, and so on). The cube's corners and edges only ever form loops of length 4 or 6, both even, and it has no triangles at all, since no three corners of a cube are all mutually joined by edges.
Conclude bipartiteness: a graph with only even length cycles is bipartite, which means its corners split cleanly into two sets with every edge crossing between the sets, and never staying inside one set.
Assign colors directly: take any corner, color it Red, then color every corner directly joined to it Blue, then every corner joined to those Red again, and so on outward. Because every cycle in the graph has even length, this alternating process never contradicts itself and finishes with a clean 2 coloring.

Since a single edge already forces its two ends to differ, at least 2 colors are always required, and the alternating process above shows 2 colors are also enough.

Let's summarize:

The cube graph has only even length cycles (4-loops on faces, 6-loops around the cube) and no odd cycles.
A graph with no odd cycles is bipartite and can always be colored with exactly 2 colors.

The minimum number of colors needed is 2, option (D).

Answer 2

Another way to find the minimum number of colors is to look directly at the cycles (closed loops of edges) in the cube graph, since the length and parity of those cycles decides the chromatic number for a graph like this one.

Look at the shortest loops: every face of the cube is a square, so walking around any one face traces a closed loop of exactly 4 edges. A loop of even length, like 4, never forces more than 2 colors on its own, since you can alternate two colors all the way around and land back on the start with no clash.
Check for odd loops: a graph needs 3 or more colors only if it contains at least one cycle with an odd number of edges (a triangle, a 5-loop, and so on). The cube's corners and edges only ever form loops of length 4 or 6, both even, and it has no triangles at all, since no three corners of a cube are all mutually joined by edges.
Conclude bipartiteness: a graph with only even length cycles is bipartite, which means its corners split cleanly into two sets with every edge crossing between the sets, and never staying inside one set.
Assign colors directly: take any corner, color it Red, then color every corner directly joined to it Blue, then every corner joined to those Red again, and so on outward. Because every cycle in the graph has even length, this alternating process never contradicts itself and finishes with a clean 2 coloring.

Since a single edge already forces its two ends to differ, at least 2 colors are always required, and the alternating process above shows 2 colors are also enough.

Let's summarize:

The cube graph has only even length cycles (4-loops on faces, 6-loops around the cube) and no odd cycles.
A graph with no odd cycles is bipartite and can always be colored with exactly 2 colors.

The minimum number of colors needed is 2, option (D).

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

Show Hint

The Correct Option is D

Solution and Explanation

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