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Understanding Euler's Path and Circuit Theorems in Graph Theory

Euler’s Path and Circuit Theorems provide insight into the properties of graphs regarding vertex parity. A connected graph with exactly two odd vertices has an Euler path, commencing at one odd vertex and concluding at the other. Conversely, if a graph has more than two odd vertices, no Euler path exists. For Euler circuits, a connected graph with all even vertices ensures at least one exists. If any vertex is odd, no circuit is possible, demonstrating the critical relationship between vertex degree and graph traversal.

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Understanding Euler's Path and Circuit Theorems in Graph Theory

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  1. Euler’s Path Theorem If a graph is connected and has exactly two odd vertices then it is a Euler path. Any such path starts with an odd vertices and ends at the other one. If a graph has more than two odd vertices it CANNOT be a Euler path. Open Unicursal (start and end is different) **Euler proved that you cannot have a graph with just 1 odd vertex.**

  2. Euler’s Circuit Theorem • If a graph is connected and every vertex is even then it has an Euler’s circuit (at least one). • If a graph has any odd vertices then it does not have an Euler’s Circuit • Closed Unicursal • Even means you have as many edges coming in as you have leaving the vertex

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