A graph needs atleast $n-1$ to remain connected. Given $ \#vertices =6$ we require atleast $5$ edges to be connected. Now having $10$ edges, we can remove $6$ edges to make the number of edges as $4$. Thus, the given graph with $4$ edges would be definitely disconnected.
Proof: The most lightly connected graph is a chain for $n$ vertices. Removing any edge from the chain breaks it. Hence $n-1$ edges are required to keep a graph connected.
Note: Just because there are $n-1$ edges, we cannot reason that the given graph is connected. Eg: A $3-vertex$ cycle and an additional isolated node has 3 edges. But still the graph is disconnected.
