G has at least one cycle or exactly one cycle?

Question

Consider a simple connected graph $G$ with n vertices and n edges $(n>2)$. Then, which of the following statements are true?

1. $G$ has no cycles
2. The graph obtained by removing any edge from $G$ is not connected
3. $G$ has at least one cycle
4. The graph obtained by removing any two edges from $G$ is not connected

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My attempt :

1. always false
2. not always true
3. true (since exactly one is subset of at least one !).
4. always true

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Can you explain in formal way, please?

Answer 1

1) Always false. A graph with no cycles (also known as a tree) has $n$ nodes and $n-1$ edges. Add one more edge and you're going to make a cycle somewhere. You can prove this by induction.

2) Not necessarily true. Easy counterexample: a graph with $n$ nodes and $n$ edges that forms a circle (i.e. a single cycle). Take out an edge anywhere and the graph is still connected.

3) Always true, see (1). If "G has no cycle" is always false, then it always has at least one cycle.

4) Always true. Think about it like this: if it's a connected graph with $n$ vertices and $n$ edges, and you remove one edge, then you have $n-1$ edges. If it's still connected, then it's a tree. (If it is not connected, then we're done.) Now you have a tree. Remove any edge from a tree, and your tree is split into two connected components.