15.2 Euler Paths & Euler Circuits

15.2 Euler Paths & Euler Circuits - vertices,...

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15.2 Euler Paths and Euler Circuits Euler’s Theorem: [1] If a graph has exactly two odd vertices, then it has at least one Euler path but no Euler circuit; each Euler path must start at one of the odd vertices and end at the other one. [2] If a graph has no odd vertices, it has at least one Euler circuit (which by definition is also a Euler path); the circuit may start and end at any vertex. [3] If a graph has more than two odd vertices, then it has no Euler paths or circuits. Fleury’s Algorithm: If Euler’s Theorem indicates the existence of a Euler path or circuit, one can be found using the following procedure: [1] If a graph has exactly two odd
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Unformatted text preview: vertices, indicating a Euler path, start at one of the odd vertices; if it has no odd vertices, indicating a Euler circuit, start at any vertex. [2] Number the edges as you trace through the graph according to the following rules: [a] After you have traveled over an edge, erase it and substitute a dashed line; [b] Travel over an edge that is a bridge only when there is no alternative. See pages 796 – 800 (4 th ed.) or 837 – 840 (5 th ed.) for the following problems: 1. Problem 6: 2. [a] Problem 8 [b] Problem 10 [c] Problem 12 3. Problem 20 4. Problem 26 5. Problems 30 6. a. Problem 42 b. Problem 44 7. Problem 48:...
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This note was uploaded on 01/04/2012 for the course MGF 1107 taught by Professor Patriciabishop during the Fall '11 term at Miami Dade College, Miami.

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