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Knowledge
The lecture provides an in-depth exploration of Euler paths and Euler circuits, defining their characteristics and conditions for existence in graphs. The presentation includes examples illustrating both concepts, proving necessary and sufficient conditions, and demonstrating Fleury’s algorithm for finding Euler circuits. Additionally, it characterizes Euler paths, highlighting the differences between paths and circuits within the context of graph theory.
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Sections
References
ch51.pdfClass Notes
Memorization
What we have learnt
- An Euler circuit visits eve...
- An Euler path visits every ...
- Graphs can have Euler circu...
Final Test
Revision Tests
What we have learnt
- An Euler circuit visits every edge of a graph exactly once and returns to the starting vertex.
- An Euler path visits every edge exactly once but does not return to the starting vertex.
- Graphs can have Euler circuits if all vertices have even degrees, or Euler paths if exactly two vertices have odd degrees.
Key Concepts
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Term: Euler Circuit
Definition: A closed trail in a graph that visits every edge once and returns to the starting vertex.
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Term: Euler Path
Definition: A trail in a graph that visits every edge once but does not return to the starting vertex.
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Term: Fleury’s Algorithm
Definition: An algorithm used to find an Euler circuit in a graph by avoiding 'cut' edges until necessary.
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Term: Even Degree
Definition: A vertex has an even degree if it is connected to an even number of edges.
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Term: Odd Degree
Definition: A vertex has an odd degree if it is connected to an odd number of edges.