All graphs to be considered are simple. A graph is matching covered if every edge of the graph belongs to a perfect matching of the graph. A graph is class 1 if it is edge 4-colorable. A graph is Hamiltonian if it has a circuit containing all its vertices. A graph is 4-regular if all its vertices have degree 4. A graph is of even order if it has an even number of vertices
I have the following question:
Question 1 Is a 4-regular, Hamiltonian graph of even order matching covered necessarily? Is there a counterexample?