Euler Path, unlike 202205101149, doesn’t need to be a cycle by itself. As long as all edges in the #202204112129 are visited exactly once, the graph is an Euler Path. Like 202205101149, the graph needs to be connected. In the graph, there must be exactly two vertices that has an odd number of edges.
A graph could be determined as an Euler Path by using #202205021959. This will result in a linear runtime complexity.