Understanding Eulerian Circuits in Graph Theory

An Eulerian circuit is a closed walk that traverses every edge of a graph exactly once and returns to the starting vertex. The concept originates from the famous Seven Bridges of Königsberg problem posed by Leonhard Euler in 1736. Euler proved that such a circuit exists only if every vertex in the graph has even degree and the graph is connected (ignoring isolated vertices). This fundamental result laid the foundation for graph theory and remains crucial in network analysis, circuit design, and combinatorial optimization.

To state it formally: Let G = (V, E) be an undirected graph. An Eulerian circuit exists if and only if every vertex v ∈ V has an even degree, and the graph is connected when considering only vertices with non‑zero degree. For directed graphs, the conditions are that every vertex has equal in‑degree and out‑degree and the underlying undirected graph is connected.

What Is Hierholzer’s Algorithm?

Hierholzer’s Algorithm, published by the German mathematician Carl Hierholzer in 1873, is an efficient method for constructing an Eulerian circuit when the necessary conditions are satisfied. It builds the circuit by finding a series of cycles and merging them. The algorithm runs in linear time O(E) with respect to the number of edges, making it optimal for dense and sparse graphs alike.

Key Concepts

  • Cycle detection: Starting from a vertex, follow unused edges until returning to the starting vertex. This forms a simple cycle.
  • Merging cycles: When a vertex on the current circuit still has unused edges, a new cycle is formed from that vertex and inserted into the circuit.
  • Edge removal: As edges are used, they are marked or removed to avoid revisiting them.

Step-by-Step Description of Hierholzer’s Algorithm

The algorithm can be implemented recursively or iteratively. The core idea is to build a circuit by repeatedly extending sub‑circuits. Below is a detailed breakdown.

Step 1: Choose a Starting Vertex

Select any vertex with at least one edge. Since the graph is connected and all degrees are even, any vertex will work. Typically the algorithm starts at vertex v.

Step 2: Traverse a Cycle

From the current vertex, follow any unused edge to a neighbor. Continue moving along unused edges, marking each edge as used, until you return to the starting vertex. This produces a cycle C. If the cycle contains all edges of the graph, the algorithm terminates – we have an Eulerian circuit.

Step 3: Find Vertices with Unused Edges

Scan the current circuit for any vertex u that still has incident unused edges. If none exist, the algorithm is complete. Otherwise, let u be such a vertex.

Step 4: Build a New Cycle from u

Starting at u, repeat the cycle‑finding process among the unused edges. This creates a new cycle C′ that begins and ends at u.

Step 5: Merge the New Cycle into the Main Circuit

Insert C′ into the main circuit at the position of u. The resulting walk is still a circuit (closed) and covers all edges visited so far. Return to Step 3.

Because every vertex has even degree, the process never gets stuck: whenever you enter a vertex, there will always be an unused edge to leave, until the vertex’s degree becomes zero. The algorithm guarantees that the final walk includes every edge exactly once.

Example: Constructing an Eulerian Circuit

Consider an undirected graph with vertices A, B, C, D, and E. Edges: AB, AC, AD, BC, BD, CE, DE. (This is a small graph where each vertex has even degree: deg(A)=3, deg(B)=3, deg(C)=2, deg(D)=3, deg(E)=1? That doesn’t satisfy even degree condition. Let’s correct: Use a graph where all degrees are even: A–B, B–C, C–D, D–A, plus A–C and B–D. That gives each vertex degree 3? That’s odd. Actually a simple even‑degree example: a triangle with each vertex degree 2? Not interesting. Let’s use a more typical example: vertices 1,2,3,4,5 with edges: 1‑2, 2‑3, 3‑1, 3‑4, 4‑5, 5‑3. Degrees: 1 (deg 2), 2 (deg 2), 3 (deg 4), 4 (deg 2), 5 (deg 2). All even, graph connected. Perfect.

Run Hierholzer’s Algorithm:

  • Start at vertex 1. Follow edges: 1‑2 (use), 2‑3 (use), now at 3. Choose unused edge 3‑4 (use), 4‑5 (use), 5‑3 (use). Return to 3, but the initial starting point was 1. We haven’t returned to 1 yet. Actually the algorithm needs to form a cycle that returns to the starting vertex. Let’s trace properly: Start at 1, go 1‑2, 2‑3, now from 3 we can go 3‑1 (unused) – that gives cycle 1‑2‑3‑1. That’s cycle C1. After that, edges left: 3‑4, 4‑5, 5‑3.
  • Scan C1: vertex 3 has unused edges. Start new cycle at 3: 3‑4, 4‑5, 5‑3. Cycle C2 = 3‑4‑5‑3.
  • Merge C2 into C1 at vertex 3: resulting circuit: 1‑2‑3‑4‑5‑3‑1. All edges used, circuit is Eulerian.

This example illustrates the elegance of the algorithm: cycles are discovered and combined seamlessly.

Complexity and Implementation Considerations

Hierholzer’s Algorithm runs in O(V + E) time when using an adjacency list representation and efficient data structures for edge removal (e.g., using iterators or linked lists). The algorithm is optimal because each edge is processed exactly once. The memory overhead is O(V + E) for storing the graph and the circuit.

For directed graphs, the same approach works provided the graph is Eulerian (in‑degree equals out‑degree at each vertex). The algorithm’s requirement of even degrees translates to the directed case as well.

Comparison with Fleury’s Algorithm

Another well‑known algorithm for finding Eulerian circuits is Fleury’s Algorithm, which works by traversing edges while ensuring that the remaining graph stays connected (i.e., avoiding bridges). Fleury’s algorithm runs in O(E2) time because it needs to check connectivity at each step. Hierholzer’s algorithm is generally preferred for its linear time complexity and simpler implementation. The only downside is that Hierholzer’s requires the graph to be Eulerian (even degrees) whereas Fleury’s can also handle semi‑Eulerian graphs (when exactly two vertices have odd degree, producing an Eulerian trail). However, Hierholzer’s can be adapted for Eulerian trails as well by adding a dummy edge between the two odd‑degree vertices, constructing the circuit, and then removing the dummy edge.

Applications of Hierholzer’s Algorithm

The ability to find an Eulerian circuit efficiently has many real‑world uses.

Chinese Postman Problem

In the Chinese Postman problem (route inspection), the goal is to find the shortest closed walk that covers every edge at least once. For graphs that are already Eulerian, the solution is simply the Eulerian circuit. Hierholzer’s algorithm provides that circuit. For non‑Eulerian graphs, the problem reduces to duplicating edges to make all degrees even, and then applying Hierholzer’s.

Network Routing and Circuit Design

Eulerian circuits are used in designing efficient routes for street sweepers, garbage collection, and network packet transmission where each link must be traversed exactly once. The algorithm helps minimize redundant travel.

DNA Fragment Assembly

In computational biology, the de Bruijn graph approach to genome assembly relies on finding Eulerian paths or circuits through k‑mer graphs. Hierholzer’s algorithm is a core component of many assemblers, enabling the reconstruction of contiguous sequences from short reads.

Computer Graphics and Maze Generation

Eulerian trails are used in generating mazes and in certain graph drawing algorithms where edges must be drawn without lifting the pen. The algorithm provides an optimal construction.

Integrated Circuit Testing

In Very Large‑Scale Integration (VLSI) design, testing all connections can be modeled as an Eulerian circuit problem, minimizing tester movement.

Further Reading and External Resources

To deepen your understanding of Eulerian circuits and Hierholzer’s algorithm, the following resources are recommended:

Conclusion

Hierholzer’s Algorithm remains a cornerstone of graph traversal for its elegance, speed, and broad applicability. By decomposing the problem into finding and merging cycles, it provides a straightforward and optimal solution for constructing Eulerian circuits. Whether you are designing network routes, assembling genomes, or solving puzzles, understanding this algorithm equips you with a powerful tool for handling graphs with even‑degree vertices. Its linear time complexity and simple recursive structure make it a favorite among algorithm enthusiasts and practitioners alike.