GATE CS 2026 Set 1 — Question 55
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Engineering Mathematics → Graphs (Data Structure) → Bipartite Detection
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Question
An undirected, unweighted, simple graph
G(V, E) is said to be 2-colorable if there exists a function such that for every .Which of the following statements about 2-colorable graphs is/are true?Correct answer
(B) If G is 2-colorable, then G may contain cycles of even length; (C) An optimal algorithm for testing whether G is 2-colorable runs in time Θ(V + E), if G is represented as an adjacency list
Solution
A graph is 2-colorable if and only if it is bipartite.
1.Statement (A): A fundamental theorem in graph theory states that a graph is bipartite if and only if it contains no odd cycles. Therefore, a 2-colorable graph cannot contain cycles of odd length. Statement (A) is false.
2.Statement (B): Bipartite graphs can certainly contain even cycles (for example, a cycle of length 4 is bipartite). Statement (B) is true.
3.Statement (C): Testing whether a graph is 2-colorable (bipartite) can be done using a graph traversal algorithm like Breadth-First Search (BFS) or Depth-First Search (DFS). In an adjacency list representation, these algorithms visit every vertex and edge once, leading to a time complexity of . This is optimal because any algorithm must at least inspect every vertex and edge. Statement (C) is true.
4.Statement (D): While an algorithm with complexity might exist, it is not the optimal time complexity because a linear time algorithm exists. Statement (D) is false.
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