Graph theory Euler graph

Question

let $G=(V,E)$ be an undirected graph wheres $deg (v) =p >1$ for all $v \in V$ and also $|V|=2p+1 ,$ I need to show that G and his Complement graph are Euler graphs....
$$$$ I think that i manage to prove that it has Hamilton path but no more progress so far... thank you for help or any direction to how to solve it

Answer 1

Every connected grapgh $G$ is Eulerian grapg iff for every vertex $v\in V_G$, $deg(v)$ is even. Suppose $G$ is not connected, because for every $v\in V_g$, $deg(v)=p$, therefore every componnet of $G$ has at least $p+1$ vertecis, and that means $V_G>2p+1$ which leads to contradiction, so $G$ is connected. Also $\sum_{v\in V_G}deg(v)=p(2p+1)=2E$, therefore $p$ is even, so $G$ is Eulerian. The same arguments works for $\bar G$, because for every $v\in V_{\bar G}$, $deg(v)=2p-p=p$.