Hi, Can i get answers to these graph theory questions below. Show that if T is a tree containing at least one vertex of degree 2, then complement T...

Hi,

Can i get answers to these graph theory questions below.

(1).    Show that if T is a tree containing at least one vertex of degree 2, then complement T  is not Eulerian.

(2).   Let G be a connected r-regular graph of order n such that complement of G is also connected. Prove that at least one of G and complement of G is Eulerian.

(3).   Let G be a connected r-regular graph of order n such that complement G is also connected. Prove that at least one of G and complement of G is Hamiltonian

(4).    Prove that for each odd integer n ≥ 3, there exists exactly one Eulerian graph of order n containing exactly three vertices of the same degree and at most two                   vertices of any other degree.