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 trip is a Hamiltonian cycle: a closed route through every city exactly once.(i) The grid has a Hamiltonian cycle, for example go along the top row, snake down and back through the remaining three rows in columns 2 to 4, and return up column 1. Label rows and columns : . So the trip is possible.(ii) There are 5 cities. Two of them, call them (top-left) and (bottom), are each joined to the same three cities , and have no other highways (each has degree 2). In a Hamiltonian cycle every degree-2 vertex must use both of its edges, so the cycle would contain all six edges . Then would have degree 3 in the cycle, which is impossible (every vertex on a cycle has degree exactly 2). Equivalently, the graph is bipartite with parts of sizes 2 and 3, and a cycle in a bipartite graph alternates sides, so it cannot cover unequal parts. So the trip is impossible.
- (A) True.
- (B), (C), (D) contradict the analysis above.
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