Ch 6 Notes - Chapter 6 Hamilton Path: a path that passes...

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Chapter 6
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Hamilton Path : a path that passes through each VERTEX of the graph once and only once Hamilton Circuit : same as a Hamilton path except that it returns to the starting point What’s the difference between HAMILTON paths/circuits and EULER paths/circuits ? Hamilton paths/circuits pass through all VERTICES Euler paths/circuits pass through all EDGES
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Top Thrill Dragster Magnum Wicked Twister Raptor Mantis Millennium Force You are at Cedar Point Amusement Park and you want to ride the roller coasters shown in the graph. The edges represent the walkways in the park and the vertices represent the roller coasters. Find a path that will take you to each roller coaster once and only once.
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E A D C B Let’s say that we want to begin at A. How many edges can we travel from A? 4 Let’s say we chose to travel to E. How many edges can we travel from E? 3 Let’s say we chose to travel to C. How many edges can we travel from C? 2 Let’s say we chose to travel to B. How many edges can we travel from B? 1 We must go to D – it’s the only vertex left and there’s only one way to get there How many different Hamilton Circuits are possible? We must go back to A to complete the circuit – there’s only one way to do that 4 • 3 • 2 • 1 • 1 • 1 = 24 4! = 24
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In general, the number of different Hamilton Circuits that are possible in a complete graph with N vertices is In the previous example, the diagram had 5 vertices. E A D C B The number of possible Hamilton Circuits was 4! (N – 1)! So if a graph has 10 vertices, it has 9! possible Hamilton Circuits One less than the number of vertices
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10 9 8 7 6 5 4 3 2 (N - 1)! N 1 2 6 24 120 720 5,040 40,320 362,880 Number of Different Possible Hamilton Circuits for N vertices 20 19 18 17 16 15 14 13 12 11 (N – 1)! N 121,645,100,408,832,000 6,402,373,705,728,000 355,687,428,096,000 20,922,789,888,000 1,307,674,368,000 87,178,291,200 6,227,020,800 479,001,600 39,916,800 3,628,800
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Las Vegas San Francisco Buffalo Myrtle Beach Fort Lauderdale It’s never too early to start thinking about spring break!
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Ch 6 Notes - Chapter 6 Hamilton Path: a path that passes...

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