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D - Make Bipartite 2 /

Time Limit: 2 sec / Memory Limit: 1024 MB

### 問題文

N 個の頂点と M 本の辺からなる単純な（すなわち、自己ループも多重辺も含まない）無向グラフ G が与えられます。 i = 1, 2, \ldots, M について、i 番目の辺は頂点 u_i と頂点 v_i を結びます。

1 \leq u \lt v \leq N を満たす整数の組 (u, v) であって、下記の 2 つの条件をともに満たすものの個数を出力してください。

• グラフ G において、頂点 u と頂点 v を結ぶ辺は存在しない。
• グラフ G に、頂点 u と頂点 v を結ぶ辺を追加して得られるグラフは、二部グラフである。

• 同じ色に塗られた頂点どうしを結ぶ辺は存在しない。

### 制約

• 2 \leq N \leq 2 \times 10^5
• 0 \leq M \leq \min \lbrace 2 \times 10^5, N(N-1)/2 \rbrace
• 1 \leq u_i, v_i \leq N
• グラフ G は単純
• 入力はすべて整数

### 入力

N M
u_1 v_1
u_2 v_2
\vdots
u_M v_M


### 入力例 1

5 4
4 2
3 1
5 2
3 2


### 出力例 1

2


### 入力例 2

4 3
3 1
3 2
1 2


### 出力例 2

0


### 入力例 3

9 11
4 9
9 1
8 2
8 3
9 2
8 4
6 7
4 6
7 5
4 5
7 8


### 出力例 3

9


Score : 400 points

### Problem Statement

You are given a simple undirected graph G with N vertices and M edges (a simple graph does not contain self-loops or multi-edges). For i = 1, 2, \ldots, M, the i-th edge connects vertex u_i and vertex v_i.

Print the number of pairs of integers (u, v) that satisfy 1 \leq u \lt v \leq N and both of the following conditions.

• The graph G does not have an edge connecting vertex u and vertex v.
• Adding an edge connecting vertex u and vertex v in the graph G results in a bipartite graph.
What is a bipartite graph?

An undirected graph is said to be bipartite if and only if one can paint each vertex black or white to satisfy the following condition.

• No edge connects vertices painted in the same color.

### Constraints

• 2 \leq N \leq 2 \times 10^5
• 0 \leq M \leq \min \lbrace 2 \times 10^5, N(N-1)/2 \rbrace
• 1 \leq u_i, v_i \leq N
• The graph G is simple.
• All values in the input are integers.

### Input

The input is given from Standard Input in the following format:

N M
u_1 v_1
u_2 v_2
\vdots
u_M v_M


### Sample Input 1

5 4
4 2
3 1
5 2
3 2


### Sample Output 1

2


We have two pairs of integers (u, v) that satisfy the conditions in the problem statement: (1, 4) and (1, 5). Thus, you should print 2.
The other pairs do not satisfy the conditions. For instance, for (1, 3), the graph G has an edge connecting vertex 1 and vertex 3; for (4, 5), adding an edge connecting vertex 4 and vertex 5 in the graph G does not result in a bipartite graph.

### Sample Input 2

4 3
3 1
3 2
1 2


### Sample Output 2

0


Note that the given graph may not be bipartite or connected.

### Sample Input 3

9 11
4 9
9 1
8 2
8 3
9 2
8 4
6 7
4 6
7 5
4 5
7 8


### Sample Output 3

9