Section Recap

You've learned Euler paths and circuits.

core concepts:

• Euler path visits every edge exactly once
• Euler circuit is an Euler path returning to start
• Undirected: $0$ or $2$ odd-degree vertices
• Directed: in-degree $=$ out-degree (with $\pm 1$ for endpoints)

Algorithm: Hierholzer's algorithm finds Euler paths in $O(E)$ time.

Applications:

• Mail delivery and routing
• De Bruijn sequences
• DNA sequencing

When you see: "visit every edge exactly once" → Euler path. Check degree conditions first, then run Hierholzer.