Eulerian Cycle Visualization
An Eulerian Cycle (or Eulerian Circuit) is a path in a graph that visits every edge exactly once and returns to the starting node. In an undirected graph, an Eulerian cycle exists if and only if every vertex has an even degree and all vertices with non-zero degree belong to a single connected component. It is closely related to the famous Seven Bridges of Königsberg problem.
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Common Interview Questions
What is the difference between an Eulerian Path and an Eulerian Cycle?
An Eulerian Path visits every edge in a graph exactly once and exists if exactly 0 or 2 vertices have an odd degree. An Eulerian Cycle is an Eulerian Path that starts and ends at the same vertex, which requires every vertex to have an even degree (0 odd degree vertices).
Why is Hierholzer's algorithm preferred over Fleury's algorithm for finding an Eulerian circuit?
Fleury's algorithm builds the circuit edge by edge, only crossing a bridge if there is no alternative. Because detecting bridges requires running DFS or BFS at each step, Fleury's runs in O(E²) time. Hierholzer's algorithm traces simple cycles and splices them together using a stack, running in optimal linear time O(V + E). Hierholzer's is therefore the universal choice in production.
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