Question

The 15 parts of the given figure are to be painted such that no two adjacent parts with shared boundaries (excluding corners) have the same color. The minimum number of colors required is:
\includegraphics[width=0.5\linewidth]{2.png}

• A. \( 4 \)
• B. \( 3 \)
• C. \( 5 \)
• D. \( 6 \)

Hint:

For graph coloring problems, convert the figure into a graph representation. The chromatic number gives the minimum number of colors required. Use logical reasoning to verify adjacency constraints.

Answer 1

The Correct Option is A

This problem is a classic example of the graph coloring problem, where the goal is to assign colors to regions such that no two adjacent regions share the same color. The "adjacent" condition applies to regions that share a boundary but not just a corner. Step 1: Analyze the figure. The given figure consists of 15 regions with shared boundaries. To solve, we map this as a graph: Each region is represented as a vertex in the graph. An edge exists between two vertices if the corresponding regions share a boundary. Step 2: Determine the chromatic number. The minimum number of colors required to color the graph such that no two adjacent vertices (regions) share the same color is known as the chromatic number. For this figure, careful analysis shows that: Regions form a complex structure, but no region is connected to more than 4 others. A minimum of 4 colors is required to ensure that no two adjacent regions have the same color. Step 3: Verify. By systematically assigning 4 colors to the regions, it is possible to satisfy the condition that no two adjacent regions share the same color. Conclusion.
The minimum number of colors required to paint the figure such that no two adjacent parts share the same color is \( 4 \), making the correct answer \( \mathbf{(1)} \).