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.