GATE CS 2024 Set 2 — Question 60
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Engineering Mathematics → Graph Theory (Math) → Graph Coloring
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Question
The chromatic number of a graph is the minimum number of colours used in a proper colouring of the graph. The chromatic number of the following graph is _________
Correct answer
2 to 2
Solution
The chromatic number of a graph is the smallest number of colors needed to color the vertices so that no two adjacent vertices share the same color.Let's analyze the structure of the given graph:
1.Vertices: It has 8 vertices, arranged as an outer square (4 vertices) and an inner square (4 vertices).
2.Edges:
- The outer vertices form a cycle of length 4 ().
- The inner vertices form a cycle of length 4 ().
- Each outer vertex is connected to two inner vertices.
- Color the outer cycle alternatingly: Top-Outer (R), Right-Outer (B), Bottom-Outer (R), Left-Outer (B). This is valid for the outer square.
- Color the inner cycle alternatingly: Top-Inner (R), Right-Inner (B), Bottom-Inner (R), Left-Inner (B). This is valid for the inner square.
- Now check the cross connections. Visually, the Top-Outer vertex (R) is connected to the Left-Inner (B) and Right-Inner (B) vertices. This is a valid connection (Red to Blue).
- Similarly, the Right-Outer vertex (B) connects to Top-Inner (R) and Bottom-Inner (R). This is valid (Blue to Red).
- By symmetry, all other connections also respect the coloring (Red connected only to Blue).
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