Eulerian path

Eulerian path
Cmglee · Public domain · source
Name	Eulerian path
Field	Graph theory
First proposed	1736
Originator	Leonhard Euler
Related	Eulerian circuit, Hamiltonian path, Königsberg bridge problem

Contents

Eulerian path An Eulerian path is a trail in a finite graph that visits every edge exactly once and may start and end at different vertices. Originating in the Königsberg bridge problem solved by Leonhard Euler, the concept influenced the development of graph theory and subsequent work by mathematicians in Switzerland, Germany, and across Europe. The study of Eulerian paths connects to problems examined by figures and institutions such as Carl Friedrich Gauss, Augustin-Louis Cauchy, Brook Taylor, University of Königsberg, École Polytechnique, and the Royal Society.

Definition

An Eulerian path in a graph is a sequence of edges forming a trail that includes each edge of the graph exactly once, distinct from a Hamiltonian path studied by William Rowan Hamilton and discussed in contexts like the Icosian game. The formalization followed work by Leonhard Euler addressing the Königsberg bridge problem, later generalized in texts from Joseph-Louis Lagrange and treated in curricula at institutions such as University of Paris and University of Göttingen. Eulerian path theory appears alongside results by Johann Carl Friedrich Gauss, Augustin-Louis Cauchy, and later expositors at Cambridge University and Princeton University.

Classical criteria for the existence of an Eulerian path were established in the 19th century and formalized in graph theory courses at Harvard University and Yale University. For finite connected graphs, necessary and sufficient conditions link vertex degrees and connectivity addressed historically by William Rowan Hamilton and refined by researchers at University of Berlin and ETH Zurich. The theorems underpinning these criteria are discussed in works by Arthur Cayley, Gustav Kirchhoff, James Joseph Sylvester, and expositions in journals such as the Proceedings of the Royal Society and publications of the American Mathematical Society.

Constructive algorithms for producing Eulerian paths were developed and implemented in computational contexts at Bell Labs, IBM Research, and university groups at Massachusetts Institute of Technology and Stanford University. Hierholzer's algorithm, with roots in 19th-century techniques, and Fleury's algorithm are standard; implementations appear in software from Microsoft Research, in libraries originating from GNU Project, and in coursework at California Institute of Technology and University of Oxford. These algorithms intersect with complexity theory explored by researchers at Clay Mathematics Institute and practical optimization studied at INRIA and Max Planck Institute.

Related concepts include the Eulerian circuit, Chinese postman problem, and arc-routing problems studied by scholars at IBM Research, Bell Labs, and in applied work tied to Daimler AG and logistics groups such as United Parcel Service. Connections to Hamiltonian paths appear in analyses by William Rowan Hamilton and optimization research at Toyota Research Institute. Extensions to directed graphs, mixed graphs, and multigraphs have been developed in collaborations involving École Polytechnique Fédérale de Lausanne, Technical University of Munich, and network research at AT&T Labs and Cisco Systems.

Eulerian paths are applied in route planning problems used by companies like DHL, FedEx, and technologies developed by Google and Uber Technologies for mapping and routing; they inform genomic assembly methods pioneered at Broad Institute and in projects at Wellcome Sanger Institute. In chemistry, Eulerian trail concepts aid studies of molecular graphs by researchers at Max Planck Institute for Chemical Energy Conversion and Royal Society of Chemistry-affiliated groups. Urban planning and transportation systems leveraging Eulerian principles have been implemented in case studies involving the New York City Department of Transportation, Transport for London, and municipal projects documented by United Nations urban programs.