Problem:
Score : 300 points

Problem Statement

You are given an undirected unweighted graph with N vertices and M edges that contains neither self-loops nor double edges.

Here, a self-loop is an edge where a_i = b_i (1≤i≤M), and double edges are two edges where (a_i,b_i)=(a_j,b_j) or (a_i,b_i)=(b_j,a_j) (1≤i<j≤M).

How many different paths start from vertex 1 and visit all the vertices exactly once?

Here, the endpoints of a path are considered visited.

For example, let us assume that the following undirected graph shown in Figure 1 is given.

Figure 1: an example of an undirected graph

The following path shown in Figure 2 satisfies the condition.

Figure 2: an example of a path that satisfies the condition

However, the following path shown in Figure 3 does not satisfy the condition, because it does not visit all the vertices.

Figure 3: an example of a path that does not satisfy the condition

Neither the following path shown in Figure 4, because it does not start from vertex 1.

Figure 4: another example of a path that does not satisfy the condition

Constraints

2≦N≦8

0≦M≦N(N-1)/2

1≦a_i<b_i≦N

The given graph contains neither self-loops nor double edges.

Input

The 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

Print the number of the different paths that start from vertex 1 and visit all the vertices exactly once.

Sample Input 1

3 3
1 2
1 3
2 3

Sample Output 1

2

The given graph is shown in the following figure:

The following two paths satisfy the condition:

Sample Input 2

7 7
1 3
2 7
3 4
4 5
4 6
5 6
6 7

Sample Output 2

1

This test case is the same as the one described in the problem statement.

Solution in C# (.NET Core 3.1.201) (judge verdict: Runtime Error)

using System;
using System.Linq;
using System.Collections.Generic;

namespace ABC
{
    class ProgramC
    {
        struct Info
        {
            public int to;
            public string path;
        }
        static Queue<Info> _q = new Queue<Info>();
        static List<int>[] _graph;
        static int _count;
        static int _n;

        static void Main(string[] args)
        {
            var inputs = Console.ReadLine().Split(" ");
            _n = int.Parse(inputs[0]);
            var m = int.Parse(inputs[1]);

            _graph = new List<int>[_n];
            for (var i = 0; i < _n; i++)
            {
                _graph[i] = new List<int>();
            }

            for (var i = 0; i < _n; i++)
            {
                var inputs1 = Console.ReadLine().Split(" ");
                var a = int.Parse(inputs1[0]) - 1;
                var b = int.Parse(inputs1[1]) - 1;
                _graph[a].Add(b);
                _graph[b].Add(a);
            }

            _q.Enqueue(new Info() { to = 0, path = "0" });
            while (_q.Count > 0)
            {
                var info = _q.Dequeue();
                var v = info.to;
                var path = info.path;

                foreach (var t in _graph[v])
                {
                    if (path.Contains(t.ToString())) continue;
                    _q.Enqueue(new Info()
                    {
                        to = t,
                        path = path + t.ToString(),
                    });
                }

                if (path.Count() == _n)
                    _count++;
            }

            Console.WriteLine(_count);
        }
    }
}