I'm trying to prove
A graph with an equal number of edges and vertices contains a cycle as a subgraph
Induction on the number of vertices:
Clearly holds for $n=3$
Assume true for all graphs on $\le n-1$ vertices.
Case 1: There is a vertex of degree 1, say $v$. Then $G = v \cup H$. By hypothesis, H contains a cycle, so G necessarily contains a cycle.
Case 2: No vertex of degree 1. So every vertex has degree $\ge 2$. Take minimum spanning tree of G, which contains no cycles and has at most $n-1$ edges. But our hypothesis says that any graph on $\le n-1$ edges contains a cycle. Contradiction.
I'm not sure about my case 2, any verification would be helpful.