GATE CS 2026 Set 2 — Question 8
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Engineering Mathematics → Graph Theory (Math) → Euler & Hamiltonian Paths
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Question
Figures (i) and (ii) represent intercity highway systems. The black dots represent cities and the line segments between them represent intercity highways.
A salesperson needs to make a trip. She needs to start from a city, visit each of the remaining cities exactly once, and finally return to the same city from which she started.
Which one of the following options is then true?
A salesperson needs to make a trip. She needs to start from a city, visit each of the remaining cities exactly once, and finally return to the same city from which she started.
Which one of the following options is then true?
Correct answer
(A) Such a trip is possible for (i), but not for (ii).
Solution
The problem asks for the existence of a Hamiltonian cycle in the given graphs. A Hamiltonian cycle is a closed path that visits every vertex of the graph exactly once and returns to the starting vertex.
1.Graph (i): This is a grid graph. A grid graph contains a Hamiltonian cycle if and only if the total number of vertices is even and . Since is even, a Hamiltonian cycle exists for graph (i).
2.Graph (ii): In this graph, the top-left vertex has a degree of 1 (it is connected only to the top-right vertex). For a Hamiltonian cycle to exist, every vertex must have a degree of at least 2, as the cycle must enter and leave each vertex through different edges. Because there is a vertex with degree 1, a Hamiltonian cycle cannot exist for graph (ii).
Thus, the trip is possible for graph (i) but not for graph (ii). The correct option is (A).Continue learning with Success Tracker
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