Graphs with both Eulerian circuits and Hamiltonian paths Which graphs have both Eulerian circuits and Hamiltonian paths, simultaneously?
Honestly I don't know the level of the question. One of my friens asked me to put the question on math.stackexchange.
Thanks.
 A: Well I think a class of graphs which fulfils your requirements are the cycle graphs. They are Hamiltonian trivially and are Eulerian because they are 2-regular.
Take a look at this page it explains the concepts and provides a few non trivial examples.
A: Actually there is no such specific class of graphs. You can always find examples that will be both Eulerian and Hamiltonian but not fit within any specification. The set of graphs you are looking for is not those compiled of cycles. 
For any $G$ with an even number of vertices the regular graph with, $$ degree(v) = \frac{n}{2} ,\frac{n}{2} +2, \frac{n}{2} + 4  \;  \;.....  \; \; or  \; \; n-1 \; \; for \; \; \forall v\in V(G) $$ will be both Euler and Hamiltonian by Dirac's Theorem.
Similarly, Given $G$ with an odd number of vertices if, $$ degree(v) = \frac{n+1}{2} ,\frac{n+1}{2} +2, \frac{n+1}{2} + 4 \; \;..... \; \; or  \; \; n \; \; for \; \; \forall v\in V(G) $$ 
then $G$ will be both Eulerian and Hamiltonian. Again the above sets do not contain all graphs which are both Hamiltonian and Euler. The main reason we cannot classify all such graphs is due to the absence of a necessary and sufficient condition for a graph to be Hamiltnonian.
