determine Eulerian or Hamiltonian
That's a very good question, and the easy answer is that checking whether a graph is Eulerian is much simpler than checking whether a graph is Hamiltonian.  You're diving head-first into the field of complexity theory and the famous question P vs NP.  (Further reading: What is the definition of P, NP, NP-complete and NP-hard.)
As you said, a graph is Eulerian if and only if the vertices have even degrees.
For checking if a graph is Hamiltonian, I could give you a &quot;certificate&quot; (or &quot;witness&quot;) if it indeed was Hamiltonian.  However, there is no anti-certificate, or a certificate for showing that the graph is non-Hamiltonian;  Checking if a graph is not Hamiltonian is a co-NP-complete problem.
In fact, we believe that any certificate for non-Hamiltonian-ness needs to be exponentially large.

To answer the question:

Try every permutation of vertices, and if one of the permutations is a cycle, then the graph is Hamiltonian.  If so, you get a certificate.
If no permutation was a cycle, the graph is not Hamiltonian.  You cannot convince your friends that the graph is non-Hamiltonian without trying all permutations*.

(Ps, using algorithmic techniques (DP) you don't have to try every permutation, but still exponentially many.)