[Math] Proving that a 4-regular graph has two edge-disjoint cycles

Question

Note: This is not an assignment/homework question, just review for an exam

We call two cycles edge-disjoint if they do not share any common edges, but they may share vertices.

Prove that any 4-regular graph contains at least two edge-disjoint cycles.

I was thinking that I could just choose two cycles, and allow them to share some vertex in order to complete the proof. But I only know that there is a single cycle in a 4-regular graph from the theorem that:

If every vertex has degree 2 or greater, then the graph contains a cycle.

I'm unclear of where to proceed.

Answer 1

If every vertex has degree 2 or greater, then the graph contains a cycle.

If graph contains a cycle then this graph contains simple cycle. Now remove all edges of a simple cycle. Remaining graph has vertices of degree 2 and 4 only. Just apply this theorem once again.