Searching for Graph (Codeforces 403A)
Tags: Others, graphs

Let's call an undirected graph of n vertices p-interesting, if the following conditions fulfill:   the graph contains exactly 2n + p edges;  the graph doesn't contain self-loops and multiple edges;  for any integer k (1 ≤ k ≤ n), any subgraph consisting of k vertices contains at most 2k + p edges. A subgraph of a graph is some set of the graph vertices and some set of the graph edges. At that, the set of edges must meet the condition: both ends of each edge from the set must belong to the chosen set of vertices. Your task is to find a p-interesting graph consisting of n vertices.

Input: The first line contains a single integer t (1 ≤ t ≤ 5) — the number of tests in the input. Next t lines each contains two space-separated integers: n, p (5 ≤ n ≤ 24; p ≥ 0; ) — the number of vertices in the graph and the interest value for the appropriate test. It is guaranteed that the required graph exists.

Output: For each of the t tests print 2n + p lines containing the description of the edges of a p-interesting graph: the i-th line must contain two space-separated integers ai, bi (1 ≤ ai, bi ≤ n; ai ≠ bi) — two vertices, connected by an edge in the resulting graph. Consider the graph vertices numbered with integers from 1 to n. Print the answers to the tests in the order the tests occur in the input. If there are multiple solutions, you can print any of them.


Sample Inputs:
1
6 0

Sample Outputs:
1 2
1 3
1 4
1 5
1 6
2 3
2 4
2 5
2 6
3 4
3 5
3 6

Solution (Python):
def main():
    t = int(input())
    for _ in range(t):
        (n, p) = map(int, input().split())
        total = 2 * n + p
        graph = [[0] * (n + 1) for _ in range(n + 1)]
        for i in range(1, n + 1):
            for j in range(1, n + 1):
                if i != j and graph[i][j] == 0:
                    print(i, j)
                    total -= 1
                    graph[i][j] = 1
                    graph[j][i] = 1
                if total == 0:
                    break
            if total == 0:
                break
if __name__ == '__main__':
    main()

Solution (Python):
def main():
    t = int(input())
    for _ in range(t):
        (n, p) = map(int, input().split())
        graph = [[0] * (n + 1) for _ in range(n + 1)]
        tota = 2 * n + p
        for i in range(1, n + 1):
            for j in range(1, n + 1):
                if i != j and graph[i][j] == 0:
                    print(i, j)
                    tota -= 1
                    graph[i][j] = 1
                    graph[j][i] = 1
                if tota == 0:
                    break
            if tota == 0:
                break
if __name__ == '__main__':
    main()

Solution (Python):
t = int(input())
for _ in range(t):
    (n, p) = list(map(int, input().split()))
    p += 2 * n
    step = 0
    result = []
    while p > 0:
        step += 1
        if p > n:
            result += [(i, (i + step) % n) for i in range(n)]
        else:
            result += [(i, (i + step) % n) for i in range(p)]
        p -= n
    result = [' '.join([str(p[0] + 1), str(p[1] + 1)]) for p in result]
    result = '\n'.join(result)
    print(result)