Eulerian path

Eulerian path is a path in a finite graph that visits every edge exactly once.
Similarly, an Eulerian circuit or Eulerian cycle is an Eulerian path that starts and ends on the same vertex.
A graph that contains a Eulerian cycle is called a Eulerian graph.

A connected graph is eulerian if and only if it is even^[every vertex of G has positive even degree].
Let $G$ be a graph, and let $v$ and $u$ be two distinct vertices of G. There is an Eulerian path from $v$ to $u$ if, and only if, $G$ is connected, $v$ and $u$ have odd degree, and all other vertices of $G$ have positive even degree.^[question 1]

Proposition: in graph that has a Eulerian cycle that is also Hamiltonian cycle, is 2-regular.
Proof: $e_{1} = (v_{0} v_{1}), e_{2} = (v_{1} v_{2}), e_{n} = (v_{n - 1} v_{n}), e_{n + 1} = (v_{n} v_{0})$ . is Eulerian and Hamiltonian cycle, and $e_{1}, \dots, e_{n + 1}$ are all the graph edges, and each one appear once time, therefore $∣ E ∣ = ∣ V ∣ = n + 1$ . now because Eulerian cycle all degere are even, and because hamiltonian cycle, there’s no vertex with 0 degree, therefore, $2∣ E ∣ = 2∣ V ∣ \leq \sum_{v \in V} deg_{G} (v) = 2∣ E ∣ ⟹ 2∣ V ∣ = \sum_{v \in V} deg_{G} (v)$ , therefore is 2-regular.
Proposition: in graph that has a Eulerian cycle that is also Hamiltonian cycle, is cycle graph. Proof: there is hamiltonian cycle, thus is connected, therefore, according to the previous proposition and question 1.2, it follows that is cycle graph.

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