Euler graph - a question about the proof I have a question about the proof of this theorem.
A graph is Eulerian $\iff$ it is connected and all its vertices have even degrees.
My question concerns "$\Leftarrow$"
Let $T=(v_0, e_1, v_1, ..., e_m, v_m)$ be a trip in Eulerian graph G=(V, E) where vertices can repeat but edges cannot. Let's consider T of the largest possible length. We prove that
(i) $v_0 = v_m$, and
(ii) $\left\{ e_i : i = 1, 2, . . . , m\right\} = E$ (but I think I understand everything about this part)
Ad (i). If $v_0 \neq v_m$ then the vertex $v_0$ is incident to an odd number
of edges of the tour $T$. But since the degree $deg_G(v_0)$ is even, there
exists an edge $e \in E(G)$ not contained in T. Hence we could extend
$T$ by this edge — a contradiction.
What I don't understand here is why $v_0$ is incident to an odd number of edges.

You have a typo: it’s when $v_0\ne v_m$ that you can conclude that $v_0$ is incident to an odd number of edges of $T$. It’s incident to $v_1$, and any other vertices of $T$ to which it is incident must come in pairs, one just before it and one just after in the tour. But then, as you say, $T$ would not be maximal, so this is impossible, and we must have $v_0=v_m$.
