Bipartite Graph Check
A medium Graph problem included in Love Babbar 450, Striver A2Z. Below: the roles whose interviews prioritise this topic, and how to practise it.
- Topic
- Graph
- Sheets
- 2
- Core for
- 11 roles
- Platform
- LeetCode
The problem
Given an undirected graph with n nodes labeled from 0 to n-1 and a list of edges, determine whether the graph is bipartite. A graph is bipartite if its nodes can be divided into two independent sets such that every edge connects a node from one set to the other.
Example 1
- Input
- n = 4, edges = [[0,1],[1,2],[2,3],[3,0]]
- Output
- true
- Why
- Nodes can be split into {0,2} and {1,3}. Every edge crosses the two sets.
Example 2
- Input
- n = 3, edges = [[0,1],[1,2],[2,0]]
- Output
- false
- Why
- A triangle cannot be split into two independent sets.
Constraints
- 1 <= n <= 100
- 0 <= edges.length <= n * (n - 1) / 2
How to think about it
Updated 2026-09-09A graph is bipartite if and only if its vertices can be 2-colored such that no edge connects two vertices of the same color. Equivalently, it contains no odd-length cycles. Pick an uncolored node, assign it color 0, and force every adjacent neighbor to color 1. Any neighbor that is already colored with your own color disproves bipartiteness on the spot.
Approaches, worst first
BFS two-coloring with queue
time O(V + E) · space O(V)
Maintain an array `color` initialized to -1. For each node 0 to n - 1 with color -1, assign color 0 and enqueue it. While the queue is non-empty, dequeue u and inspect neighbors v: if uncolored, set `color[v] = 1 - color[u]` and enqueue; if already matching `color[u]`, return false.
DFS two-coloring recursionWrite this one
time O(V + E) · space O(V)
Use recursive coloring: `dfs(node, c)` paints the node with c and calls `dfs(neighbor, 1 - c)`. If a neighbor is already colored with c, abort immediately with false.
Where people lose marks · 3
- Graph may be disconnected; looping over all nodes 0 to n - 1 is mandatory, otherwise disconnected odd cycles are never discovered.
- Self-loops `(u, u)` automatically make a graph non-bipartite because a vertex would have to share an edge with itself.
- Initializing color array with 0 instead of -1 collides with the first valid color choice 0.
The theory behind it
Graph — the ground this problem stands on. All Graph problems
What Graph is
A graph is a network of individual points, called vertices or nodes, connected by lines called edges. Think of a subway transit map, an electrical circuit, or a web of social friends. Unlike a tree, a graph has no designated top node and no parent-child hierarchy. Connections can run one-way or both ways, and paths can loop back on themselves to form closed cycles.
When to reach for it
Reach for graph algorithms when inputs describe relationships, networks, flights between cities, course prerequisites, or clone networks. Signals include finding the shortest route across unweighted connections, ordering tasks that depend on earlier tasks, counting isolated clusters, or checking whether a path contains an infinite loop. Whenever problems present pairs of related entities and ask for reachability, distances, or dependencies, graph representations apply.
How the pattern works
First convert edge lists into an adjacency list, mapping each node to an array of its neighbors. Choose your exploration strategy based on the goal: use a queue and breadth-first search to find the shortest path in unweighted networks, or use recursion and depth-first search to explore full paths and detect cycles. Because graphs can have loops, always track visited nodes in a set or boolean array. Add nodes to the visited set at the moment they enter the queue so they are never visited twice.
What each operation costs
| Operation | Time |
|---|---|
| visit all nodes and edges via search | O(v + e) |
| topological sort using in-degree counts | O(v + e) |
| shortest path using dijkstra with a min-heap | O((v + e) log v) |
What usually goes wrong with Graph
- Adding a node to the visited set when popping from the queue instead of when pushing, which lets neighboring nodes enqueue duplicate entries and wastes memory.
- Failing to check for cycles in directed graphs when finding prerequisite orders, causing topological sort routines to hang or return incomplete lists.
- Assuming an input graph is fully connected and scanning from only a single starting node, missing disconnected islands and isolated components.
Which roles need this problem
Graph is a core topic for these 11 roles — if you're targeting one of them, this problem is early in your path, not optional.
Secondary for 6 more roles, including Performance Engineer, Search Engineer, Information Retrieval Engineer.
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