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

Below are the $5$ non-isomorphic $11$-vertex $52$-edge graphs.  The three non-edges are in blue, and a Hamilton cycle is highlighted in orange in each case.  The vertices are ascribed their degrees.  This is sufficient to determine the existence of an Eulerian trail and Eulerian circuit in each case using the properties:


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*An undirected graph has an Eulerian trail if and only if at most two vertices have odd degree, and if all of its vertices with nonzero degree belong to a single connected component.


*An undirected graph has an Eulerian cycle if and only if every vertex has even degree, and all of its vertices with nonzero degree belong to a single connected component.
(Source: Wikipedia.)


