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. The concept is named after Leonhard Euler, a renowned Swiss mathematician who proved the existence of such paths in a network with a specific set of properties in the 18th century.
To determine whether a graph has an Euler path or not, we use the following criteria:
1. A graph has an Euler path if and only if it is connected and has 0 or 2 vertices with odd degrees (number of edges incident on a vertex).
2. If a graph has 0 vertices with odd degrees, then it has an Euler circuit (an Euler path that starts and ends at the same vertex).
3. If a graph has 2 vertices with odd degrees, then it has an Euler path that starts at one of the vertices with odd degree and ends at the other.
4. If a graph has more than 2 vertices with odd degrees, it does not have an Euler path or circuit.
To find an Euler path in a graph, we can use the Fleury’s algorithm which is a step-by-step process that systematically takes edges which result in the traversal of edges in graph, so that it can find an Eulerian path/circuit.
In summary, an Euler path is a path in a graph which visits each edge exactly once, and a graph has an Euler path/circuit if and only if it has the specific properties such as having 0 or 2 vertices with odd degrees.
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