Let $G=(V,E)$ be a graph that can be partitioned into Hamiltonian cycles. Show that there is a Eulerian cycle in $G$.
My intuition: I need help with the proof (I'm not sure my intuition is right) taking the union of all the subsets gives us a Hamiltonian cycle in $G$ Hamiltonian cycle has no repetitions when it comes to vertices then if there are no repetitions it means that there is no repetition when it comes to the edges either which means there has to be a Eulerian cycle by defintion.
Thanks in advance
For a given Hamiltonian cycle, every vertex is incident to two edges in it. Since the graph can be partitioned into such cycles, every vertex must have the same even degree, and so it must have an Eulerian cycle. (The other condition for an Eulerian cycle, connectedness, is satisfied because there is a Hamiltonian cycle.)