TITLE: how to construct a closed trail starting at a given vertex and show it exists in an eulerian graph
QUESTION [1 upvotes]: Problem: Let $G$ be an eulerian graph.
$(a)$ Explain in detail how to construct a closed trail starting at a given vertex $x$ of $G$.
Prove that such a trail exists for every $x \in V (G)$ (the vertice set of $G$).
My attempt:
a) First order the vertices of the graph, say $\{v_1,...,v_n\}$. Next I would proceed in order so that if there is an edge connecting $v_i, v_j$ where $j=i+1, 1\leq i \leq n$ then add it to the trail, otherwise continue for $j=i+2$, etc. until you have a path to the $v_n$, and then return backwards doing the same thing but for edges not added to the path where now $v_i, v_j$ form an edge with
$j=i-1$ for  $i-j<n-1$ until you return to $v_1$.
I am not sure how to show that such a trail exists however. Any hints appreciated.
Edit: My definitions from my notes:
An Euler trail in a graph is a trail that contains every edge of the graph.
An Euler tour is a closed Euler trail. A graph is called eulerian is it has an Euler tour

REPLY [1 votes]: Sketch of solution (the following contains a few claims that need to be proven explicitly):
If the graph is Eulerian, then note that if you just start at $x$ and walk an arbitrary path, you must eventually end up back at $x$ at some point, making a (not necessarily Eulerian) loop.
Next, if the path is Eulerian, then rejoice, for you are done. If not, then there must be some vertex $y$ along the chosen path that has unused edges. Start at $y$, and along the same lines as the paragraph above, make a loop that starts and ends at $y$, using only edges that wasn't used by the original $x$-loop. Now take the original loop, when you get to $y$, do the $y$-loop, then continue along the original loop after that. You have now created a longer loop.
You can keep repeating the above paragraph as long as the path you have so far isn't Eulerian.
{"set_name": "stack_exchange", "score": 1, "question_id": 2160101}