Check if a graph is Eulerian 
Let $G=((2,3,4,5,6,7),E)$ be a graph such that {$x$,$y$} $\in E$ if and only if the product of $x$ and $y$ is even, decide if G is an Eulerian graph.

My attempt
I tried to plot the graph, this is the result:

So, if my deductions are true, this is not an Eulerian graph because it's connected but all the vertices doesn't have an even degree. For example $deg(2)=5$. Moreover, there is no trace of Eulerian trails.
I cannot figure out if this assumptions are presumably correct.

You don't need to draw the graph --- you can easily do this analytically.
You have 3 odd-numbered vertices and 3 even-numbered vertices. A product $xy$ is even iff at least one of $x,y$ is even. 
A graph has an eulerian cycle iff every vertex is of even degree. So take an odd-numbered vertex, e.g. 3. It will have an even product with all the even-numbered vertices, so it has 3 edges to even vertices. It will have an odd product with the odd vertices, so it does not have any edges to any odd-numbered vertices. So $3$ has an odd degree, violating the necessary condition for an Eulerian graph. So G is not Eulerian.
