GATE CS 2018 Set 1 — Question 28
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Engineering Mathematics → Graph Theory (Math) → Graph Coloring
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Question
The chromatic number of the following graph is _______.
Correct answer
3 to 3
Solution
To find the chromatic number of the given graph, we first identify all the edges from the visual representation:
.
.
1.Lower Bound: The graph contains several cliques of size 3 (triangles), such as , , , and . Since a requires at least 3 colors to ensure no two adjacent vertices share the same color, we have .
2.Upper Bound: We can demonstrate that the graph is 3-colorable by providing a valid assignment of 3 colors (say 1, 2, and 3):
- Color 1:
- Color 2:
- Color 3:
- connects colors 2 and 3 (Valid)
- connects colors 2 and 1 (Valid)
- connects colors 2 and 3 (Valid)
- connects colors 3 and 1 (Valid)
- connects colors 3 and 1 (Valid)
- connects colors 1 and 2 (Valid)
- connects colors 1 and 3 (Valid)
- connects colors 2 and 3 (Valid)
- connects colors 2 and 1 (Valid)
- connects colors 3 and 1 (Valid)
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