GATE CS 2019 Set 1 — Question 22
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Engineering Mathematics → Graph Theory (Math) → Euler & Hamiltonian Paths
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Question
Let be an undirected complete graph on vertices, where . Then, the number of different Hamiltonian cycles in is equal to
Correct answer
(C OR D)
Solution
For a complete graph with labeled vertices:
1.The total number of permutations of vertices is .
2.A Hamiltonian cycle visits every vertex exactly once and returns to the start. Since the starting vertex can be any of the vertices without changing the cycle, we divide by : .
3.Since the graph is undirected, traversing the cycle in clockwise or anti-clockwise direction results in the same set of edges. Thus, we divide by 2.
The number of distinct Hamiltonian cycles is .Continue learning with Success Tracker
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