Property of a connected graph with $\text{deg(all nodes)}=\text{even}$ In my lecture nodes it is proposed that for a connected graph $G$ with the degree of all nodes being even, there exist two paths between any two nodes $x,y\in G$ with no common edges. From the very definition of a connected graph, there must exist one path between $x$ and $y$; the reason why there must exist two paths is then because we can simply "walk in the opposite direction" to get a new path? Also, how can it be shown that the two paths considered contain no common edges?

Since all degrees are even and the graph is connected, it has an Eulerian circuit. Starting at $x$ and walking clockwise around the Eulerian circuit gives one trail from $x$ to $y$. If this trail has repeated vertices, they can be "short-circuited" until only a path remains.
Starting at $x$ and walking counter-clockwise gives a different path from $x$ to $y$, with no edges in common with the first, because all edges in the circuit are distinct.
