Industrial and Urban Applications of Eulerian and Chinese Walks

Abstract

Eulerian walks are paths that visit each edge once in a connected graph. When the extremities of the walk are confused, then it is called Eulerian cycle or closed Eulerian walk. Introduced by Euler in 1736, Eulerian cycle concept was the historical beginning of the Graph theory. On account of the difficulty to get an Eulerian walk in a nonspecific graph, many problems were formulated with the aim to find a “weak Eulerian” walk. Thus, Chinese walk and cycle concept appeared consisting of visiting all edges of a connected graph at least one time. This concept was introduced due to the Kwan studies for postman problem. Furthermore, as each graph admits a Chinese walk, and a cycle respectively, postman studies has been of use in many modeling formulations and has given a wide range of applications relevant to transportation, urban planning, and industrial manufacturing among others. Laser or water cutting presents a technology for industrial manufacturing consisting of using water or laser to cut (metallic) materials for producing tools where it is plausible to follow Chinese walks to ensure efficiency.