Let us consider a graph G(n,e) where each player represents a vertex (n) and each match represent an edge (e).
There exists an edge between 2 vertices if there is a match scheduled between the 2 players.
The problem is to make sure that the graph G is connected. We need to determine minimum number of edges needed so that the graph G is connected.
Solution:
Worst case: 1 vertex is not connected.
Maximum number of edges possible in such case is when the rest of the vertices (n-1) form a complete graph.
Here, edges possible = (n-1)C2 = ((n-1)*(n-2))/2
Now, if we add just 1 more edge to this number, the vertex which was isolated will get connected.
Hence, e > ((n-1)*(n-2)) / 2 to make sure no player is left isolated.
Hope this helps!