We have a theorem : " A connected grapf G is an Eular graph if and only if all vertices of G are of even degree". and we know a Eular graph has an Eular path.
Observe that, in the given graph, deg(1) = 2,deg(2) =3,deg(3)=2, = deg(4) = 3, deg(5) = 5, deg(6) =3. ∴ By the stated theorem, the given graph is not an Eular graph and it does n't have any Eular path.