GATE CS 2026 Set 1 — Question 57
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Engineering Mathematics → Graph Theory (Math) → Matching
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Question
Let be an undirected graph, which is a path on 8 vertices. The number of matchings in is ______. (answer in integer)
Correct answer
34 to 34
Solution
The number of matchings in a path graph (a graph with vertices and edges) can be determined using a recurrence relation. Let be the number of matchings in a path with vertices.Step 1: Base Cases
Consider a path with vertices and edges where . A matching either contains the last edge or it does not.
Starting from the base cases, we compute the sequence (which is the Fibonacci sequence shifted):
- For (1 vertex, 0 edges): The only matching is the empty set . Thus, .
- For (2 vertices, 1 edge ): The matchings are and . Thus, .
Consider a path with vertices and edges where . A matching either contains the last edge or it does not.
- Case 1: . In this case, must be a matching of the path formed by the first vertices. There are such matchings.
- Case 2: . If is in the matching, then the edge cannot be in because it shares vertex with . The remaining edges must form a matching of the path formed by the first vertices. There are such matchings.
Starting from the base cases, we compute the sequence (which is the Fibonacci sequence shifted):
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