# Does this graph have Hamiltonian path and/or Eulerian paths?

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.

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 $\operatorname{deg}v_0=1$).