Problem:
Score : 500 points

Problem Statement

Rng has a connected undirected graph with N vertices.
Currently, there are M edges in the graph, and the i-th edge connects Vertices A_i and B_i.

Rng will add new edges to the graph by repeating the following operation:

Operation: Choose u and v (u \neq v) such that Vertex v can be reached by traversing exactly three edges from Vertex u, and add an edge connecting Vertices u and v. It is not allowed to add an edge if there is already an edge connecting Vertices u and v.

Find the maximum possible number of edges that can be added.

Constraints

2 \leq N \leq 10^5

1 \leq M \leq 10^5

1 \leq A_i,B_i \leq N

The graph has no self-loops or multiple edges.

The graph is connected.

Input

Input is given from Standard Input in the following format:

N M
A_1 B_1
A_2 B_2
:
A_M B_M

Output

Find the maximum possible number of edges that can be added.

Sample Input 1

6 5
1 2
2 3
3 4
4 5
5 6

Sample Output 1

4

If we add edges as shown below, four edges can be added, and no more.

Sample Input 2

5 5
1 2
2 3
3 1
5 4
5 1

Sample Output 2

5

Five edges can be added, for example, as follows:

Add an edge connecting Vertex 5 and Vertex 3.

Add an edge connecting Vertex 5 and Vertex 2.

Add an edge connecting Vertex 4 and Vertex 1.

Add an edge connecting Vertex 4 and Vertex 2.

Add an edge connecting Vertex 4 and Vertex 3.

Solution in Java7 (OpenJDK 1.7.0) (judge verdict: Wrong Answer)

import java.io.*;
import java.util.*;

public class Main {
	static List<Integer>[] graph;
	static int[] col;
	
	public static void main(String[] args) {
		StdIn in = new StdIn();
		int n=in.nextInt(), m=in.nextInt();
		graph = new List[n];
		for(int i=0; i<n; ++i)
			graph[i] = new ArrayList<Integer>();
		for(int i=0; i<m; ++i) {
			int a=in.nextInt()-1, b=in.nextInt()-1;
			graph[a].add(b);
			graph[b].add(a);
		}
		boolean mag1 = false;
		for(int i=0; i<n; ++i) {
			if(graph[i].size()>1) {
				if(mag1) {
					mag1=false;
					break;
				}
				mag1=true;
			}
		}
		if(mag1) {
			System.out.println(0);
			return;
		}
		col = new int[n];
		Arrays.fill(col, -1);
		boolean hasOddCycle = dfs1(0, 0);
		if(hasOddCycle) {
			System.out.println((long)n*(n-1)/2-m);
		} else {
			System.out.println((long)n*n/4-m);
		}
	}
	
	static boolean dfs1(int node, int cCol) {
		col[node]=cCol;
		for(int neighbor : graph[node]) {
			if(col[neighbor]==-1&&dfs1(neighbor, cCol^1))
				return true;
			if(col[neighbor]==cCol)
				return true;
		}
		return false;
	}
	
	interface Input {
		public String next();
		public String nextLine();
		public int nextInt();
		public long nextLong();
		public double nextDouble();
	}
	static class StdIn implements Input {
		final private int BUFFER_SIZE = 1 << 16;
		private DataInputStream din;
		private byte[] buffer;
		private int bufferPointer, bytesRead;

		public StdIn() {
			din = new DataInputStream(System.in);
			buffer = new byte[BUFFER_SIZE];
			bufferPointer = bytesRead = 0;
		}
		
		public StdIn(String filename) {
			try{
				din = new DataInputStream(new FileInputStream(filename));
			} catch(Exception e) {
				throw new RuntimeException();
			}
			buffer = new byte[BUFFER_SIZE];
			bufferPointer = bytesRead = 0;
		}
		
		public String next() {
			int c;
			while((c=read())!=-1&&(c==' '||c=='\n'||c=='\r'));
			StringBuilder s = new StringBuilder();
			while (c != -1)
			{
				if (c == ' ' || c == '\n'||c=='\r')
					break;
				s.append((char)c);
				c=read();
			}
			return s.toString();
		}

		public String nextLine() {
			int c;
			while((c=read())!=-1&&(c==' '||c=='\n'||c=='\r'));
			StringBuilder s = new StringBuilder();
			while (c != -1)
			{
				if (c == '\n'||c=='\r')
					break;
				s.append((char)c);
				c = read();
			}
			return s.toString();
		}

		public int nextInt() {
			int ret = 0;
			byte c = read();
			while (c <= ' ')
				c = read();
			boolean neg = (c == '-');
			if (neg)
				c = read();
			do
			{
				ret = ret * 10 + c - '0';
			}  while ((c = read()) >= '0' && c <= '9');

			if (neg)
				return -ret;
			return ret;
		}
		
		public int[] readIntArray(int n) {
			int[] ar = new int[n];
			for(int i=0; i<n; ++i)
				ar[i]=nextInt();
			return ar;
		}

		public long nextLong() {
			long ret = 0;
			byte c = read();
			while (c <= ' ')
				c = read();
			boolean neg = (c == '-');
			if (neg)
				c = read();
			do {
				ret = ret * 10 + c - '0';
			}
			while ((c = read()) >= '0' && c <= '9');
			if (neg)
				return -ret;
			return ret;
		}
		
		public long[] readLongArray(int n) {
			long[] ar = new long[n];
			for(int i=0; i<n; ++i)
				ar[i]=nextLong();
			return ar;
		}

		public double nextDouble() {
			double ret = 0, div = 1;
			byte c = read();
			while (c <= ' ')
				c = read();
			boolean neg = (c == '-');
			if (neg)
				c = read();

			do {
				ret = ret * 10 + c - '0';
			}
			while ((c = read()) >= '0' && c <= '9');

			if (c == '.')
			{
				while ((c = read()) >= '0' && c <= '9')
				{
					ret += (c - '0') / (div *= 10);
				}
			}

			if (neg)
				return -ret;
			return ret;
		}

		private void fillBuffer() throws IOException {
			bytesRead = din.read(buffer, bufferPointer = 0, BUFFER_SIZE);
			if (bytesRead == -1)
				buffer[0] = -1;
		}

		private byte read() {
			try{
				if (bufferPointer == bytesRead)
					fillBuffer();
				return buffer[bufferPointer++];
			} catch(IOException e) {
				throw new RuntimeException();
			}
		}

		public void close() throws IOException {
			if (din == null)
				return;
			din.close();
		}
	}
}