Hierholzer’s Algorithm: From Existence to Construction

1. Introduction

Graphs model networks everywhere—roads, circuits, social ties. An Eulerian trail uses every edge exactly once; if it ends where it started, it’s an Eulerian circuit. Euler’s 1736 Königsberg bridges posed the first existence question. In 1873, Hierholzer gave a linear-time construction.

This blog covers

• Basic Graph Definitions
• Eulerian Concepts
• Existence Theorems
• Supporting Lemmas
• Hierholzer’s Algorithm
• Worked Examples
• Visualization & Animations
• Edge Cases and Pitfalls
• Algorithm Implementation
• Applications of Eulerian Paths
• Historical Context & Origins
• Further Reading & References