# 785. Is Graph Bipartite?

#### Medium

***

There is an **undirected** graph with `n` nodes, where each node is numbered between `0` and `n - 1`. You are given a 2D array `graph`, where `graph[u]` is an array of nodes that node `u` is adjacent to. More formally, for each `v` in `graph[u]`, there is an undirected edge between node `u` and node `v`. The graph has the following properties:

* There are no self-edges (`graph[u]` does not contain `u`).
* There are no parallel edges (`graph[u]` does not contain duplicate values).
* If `v` is in `graph[u]`, then `u` is in `graph[v]` (the graph is undirected).
* The graph may not be connected, meaning there may be two nodes `u` and `v` such that there is no path between them.

A graph is **bipartite** if the nodes can be partitioned into two independent sets `A` and `B` such that **every** edge in the graph connects a node in set `A` and a node in set `B`.

Return `true` *if and only if it is **bipartite***.

 

**Example 1:**

![](https://assets.leetcode.com/uploads/2020/10/21/bi2.jpg)

```
Input: graph = [[1,2,3],[0,2],[0,1,3],[0,2]]
Output: false
Explanation: There is no way to partition the nodes into two independent sets such that every edge connects a node in one and a node in the other.
```

**Example 2:**

![](https://assets.leetcode.com/uploads/2020/10/21/bi1.jpg)

```
Input: graph = [[1,3],[0,2],[1,3],[0,2]]
Output: true
Explanation: We can partition the nodes into two sets: {0, 2} and {1, 3}.
```

 

**Constraints:**

* `graph.length == n`
* `1 <= n <= 100`
* `0 <= graph[u].length < n`
* `0 <= graph[u][i] <= n - 1`
* `graph[u]` does not contain `u`.
* All the values of `graph[u]` are **unique**.
* If `graph[u]` contains `v`, then `graph[v]` contains `u`.

#### Approach :

A succesful completion of the 2-coloring of a bipartite graph will look like the following:

[![d1](https://image.ibb.co/mGTTCc/d1.jpg)](https://imgbb.com/)

If at any point, we find that the node we are about to color with Yellow is already colored with Blue (or vice versa), this essentially means that the following non-bipartiteness exists:

[![d2](https://image.ibb.co/i3o6yH/d2.jpg)](https://imgbb.com/)

```python
class Solution:
    def isBipartite(self, graph: List[List[int]]) -> bool:
        self.graph = graph
        # Graph Coloring
        self.color = {}
        for index in range(len(graph)):
            if index not in self.color:
                # Initial Color of node is 0
                self.color[index] = 0
                # Call for DFS Traversal of this node
                if not self.dfs(index):
                    return False
        return True
        
    def dfs(self, index):
        # Traverse Adjacent Nodes
        for adj in self.graph[index]:
            # If Adjacent node is not colored yet, color it
            if adj not in self.color:
                self.color[adj] = 1 - self.color[index]
                # Call for this adjacent node DFS
                if not self.dfs(adj):
                    return False
            else:
                # If adjacent node is colored and having same color as node on other edge then 
                # it means they are in same setso return False
                if self.color[adj] == self.color[index]:
                    return False
        return True
        
```