For this graph, do Hamiltonian and Eulerian paths exist or not?
Basic definition A cycle that uses every vertex in a graph exactly once is called a Hamilton cycle, and a path that uses every vertex in a graph exactly once is called a Hamilton path.
So I think it follows Hamilton path, is it correct?
EDIT:
To check whether these graph follows Eulerian circuit:
A Euler circuit is a circuit that uses every edge of a graph
exactly once. A Euler circuit starts and ends at the same vertex.
Best Answer
Recall that an Eulerian path exists iff there are exactly zero or two odd vertices. Since $v_0$, $v_2$, $v_4$, and $v_5$ have odd degree, there is no Eulerian path in the first graph.
It is clear from inspection that the first graph admits a Hamiltonian path but no Hamiltonian cycle (since $\operatorname{deg}v_0=1$).
The other two graphs posted each have exactly two odd vertices, and so admit an Eulerian path but not an Eulerian circuit.