3-5 vertices, simple graphs. Learn the basics of Eulerian paths.
5-8 vertices, multigraphs, real-world contexts. Harder route-finding.
6-12 vertices, complex multigraphs, circuit boards, delivery routes. Full Euler analysis.
An Eulerian path (crossing every edge exactly once) exists if and only if the graph has 0 or 2 vertices with an odd number of edges (odd degree).
0 odd-degree vertices = Eulerian circuit (start and end at the same vertex).
2 odd-degree vertices = Eulerian path (must start at one odd vertex and end at the other).
4 or more odd-degree vertices = impossible.