Euler path
a route that covers every edge in graph exactly once
An Euler path is a path in a graph that visits each edge exactly once. A graph has an Euler path if and only if there are either one or two vertices of odd degree. If there are two vertices of odd degree, then the Euler path must start at one of those vertices and end at the other. If there is only one vertex of odd degree, then the Euler path must start or end at that vertex.
To find an Euler path, we typically use the Hierholzer’s algorithm, which is as follows:
1. Choose a starting vertex and follow a path until all edges have been visited.
2. If we end up at the starting vertex and all edges have been visited, we have found an Euler cycle. Otherwise, if there are still unvisited edges, choose a vertex on the path that has unvisited edges and continue until all edges have been visited.
3. Reverse the path as necessary to get the Euler path.
For example, consider the following graph:
“`
1—2—3—4
|\ | | /|
| \ | | / |
| \| |/ |
5—6—7—8
“`
This graph has vertices 1, 5, 7, and 8 with odd degree, so it has an Euler path. We can start at vertex 1 and follow a path until we reach vertex 2. At vertex 2, we can follow a path until we reach vertex 5. At vertex 5, we can follow a path until we reach vertex 6. At vertex 6, we can follow a path until we reach vertex 7. At vertex 7, we can follow a path until we reach vertex 8. At vertex 8, we can follow a path until we reach vertex 3. At vertex 3, we can follow a path until we reach vertex 4. At vertex 4, we can follow a path until we reach vertex 2. At vertex 2, we have visited all edges but have not returned to the starting vertex, so we can choose vertex 1 as the next starting vertex. We can then follow a path from vertex 1 to vertex 2, and then follow the same path as before to get the Euler path: 1-2-6-5-2-3-4-7-8.
Note that if a graph has no Euler path, it does not necessarily mean that it cannot be traversed in some other way. Also, Euler paths can be used to solve a variety of problems in computer science, such as network routing and DNA sequencing.
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