Does a path can be Hamiltonian and Eulerian at the same time?

Question

If so does it force it to be a simple circle?

Or any other restrictions?

How would it look like?

Thanks in advance

Answer 1

A path is Hamiltonian if each vertex is visited exactly once. A path is Eulerian if every edge is traversed exactly once. Clearly, these conditions are not mutually exclusive for all graphs: if a simple connected graph $G$ itself consists of a path (so exactly two vertices have degree $1$ and all other vertices have degree $2$), then that path is both Hamiltonian and Eulerian. If $G$ is a cycle, then that cycle is Hamiltonian and Eulerian.

Answer 2

To set the record clear: Yes.

A Path can be both Eularian and Hamiltonian. A Hamiltonian path is a spanning path, and an Eularian path goes through each edge exactly once. To consider a path holding both properties at the same time, think of the maximal path in $P_n$.

Answer 3

The answer is yes.

We have two criteria to meet:

$1.$ Have either $2$ odd vertices or have none at all.

$2.$ Travel to each vertex once and only once, and return to the starting point.

These can both be met, and an example is:

(Source: "Eulerian and Hamiltonian Graphs", Mathematics Learning Center, p.3)

Note: I have assumed for criteria $2$ that you are referring to a Hamiltonian circuit.