Prove that connected graph G, with 11 vertices and and 52 edges, is Hamiltonian Is this graph always, sometimes, or never Eulerian? Give a proof or a pair of examples to justify your answer
Could G contain an Euler trail? Must G contain an Euler trail? Fully justify your answer

$G$ is obtained from $K_{11}$ by removing three edges $e_1,e_2,e_3$. We label now the vertices of $G$ the following way:
$$e_1=(1,3)$$
Label the unlabeled vertices of $e_2$ by the smallest unused odd numbers, and label the unlabeled vertices of $e_3$ by the smallest unused odd numbers. Note that by our choices $e_3 \neq (1,11)$, since if $e_3$ uses the vertex $1$, the remaining vertex is labeled by a number $\leq 9$.
Label the remaining vertices some random way.
Then $1-2-3-4-5-6-7-8-9-10-11-1$ is an Hamiltonian cycle in $G$. 
