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Exploring Hamilton Circuits in Complete Graphs

Delve into the intricacies of Hamilton circuits in complete graphs, unraveling the diverse paths and multiple arrangements within. Discover the abundance of Hamilton circuits in K4 and K5, showcasing the numerous ways they can be arranged and visualized through detailed examples.

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Exploring Hamilton Circuits in Complete Graphs

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  1. 6 The Mathematics of Touring 6.1 Hamilton Paths and Hamilton Circuits 6.2 Complete Graphs? 6.3 Traveling Salesman Problems 6.4 Simple Strategies for Solving TSPs 6.5 The Brute-Force and Nearest-Neighbor Algorithms 6.6 Approximate Algorithms 6.7 The Repetitive Nearest-Neighbor Algorithm 6.8 The Cheapest-Link Algorithm

  2. How Many Hamilton Circuits? Sometimes the question, Does the graph have a Hamilton circuit? has an obviousyes answer, and the more relevant question turns out to be, How many differentHamilton circuits does it have? In this section we will answer this question for animportant family of graphs called complete graphs.

  3. How Many Hamilton Circuits? One of the key properties of KN is that every vertex has degree N – 1. Thisimplies that the sum of the degrees of all the vertices is N(N – 1), and it followsfrom Euler’s sum of degrees theorem that the number of edges in KNis N(N – 1)/2. For a graph with N vertices and no multiple edges or loops,N(N – 1)/2is the maximum number of edges possible, and this maximum canonly occur when the graph is KN.

  4. NUMBER OF EDGES IN KN ■KNhas N(N – 1)/2edges. ■Of all graphs with N vertices and no multiple edges or loops, KN has the most edges. Because KN has a complete set of edges (every vertex is connected to everyother vertex), it also has a complete set of Hamilton circuits –you can travel thevertices in any sequence you choose and you will not get stuck.

  5. Example 6.2 Hamilton Circuits in K4 If we travel the four vertices of K4 in an arbitrary order, we get a Hamilton path.For example, C, A,D,B is a Hamilton path [Fig. (a)]; D,C, A,B is another one[Fig. (b)]; and so on.

  6. Example 6.2 Hamilton Circuits in K4 Each of these Hamilton paths can be closed into a Hamiltoncircuit–the path C, A, D, B begets the circuit C, A, D, B, C [Fig. (c)]; the pathD, C, A, B begets the circuit D, C, A, B, D [Fig. (d)]; and so on.

  7. Example 6.2 Hamilton Circuits in K4 It looks like wehave an abundance of Hamilton circuits, but it is important to remember that thesame Hamilton circuit can be written in many ways. For example,C,A,D,B,Cis the same circuit as A,D,B,C,A –the figure describes either one–the only difference is that in the first case we used C as the reference point; in the secondcase we used A as the reference point.

  8. Example 6.2 Hamilton Circuits in K4 There are two additional sequences thatdescribe this same Hamilton circuit: D, B, C, A, D (with reference point D) andB, C, A, D, B (with reference point B). Taking all this into account, there are six different Hamilton circuits in K4, as shown in the Table on the next slide (the table also shows thefour different ways each circuit can be written).

  9. Example 6.2 Hamilton Circuits in K4

  10. Example 6.2 Hamilton Circuits in K5 Let’s try to list all the Hamilton circuits in K5. For simplicity, we will write eachcircuit just once, using a common reference point – say A. (As long as we are consistent, it doesn’t really matter which reference point wepick.) Each of the Hamilton circuits will be described bya sequence that starts and ends with A, with the letters B,C, D, and E sandwiched in between in some order. Thereare 4  3  2  1 = 24 different ways to shuffle the letters B, C, D, and E, each producing a different Hamiltoncircuit.

  11. Example 6.2 Hamilton Circuits in K5 The complete list of the 24 Hamilton circuits in K5 is shown in the table on the next slide. The table is laid out so that each ofthe circuits in the table is directly opposite its mirror-image circuit(the circuit with vertices listed in reverseorder). Although they are close relatives, a circuit and itsmirror image are not considered the same circuit.

  12. Example 6.2 Hamilton Circuits in K5 The complete list of the 24 Hamilton circuits in K5.

  13. Example 6.2 Hamilton Circuits in K5 Here are three of the 24 Hamilton circuits in K5.

  14. Number of Hamilton Circuits “What is the number of Hamilton circuits in KN?” boils down to theequivalent question, “How many different ways are there to rearrange the (N – 1) vertices? The answer isgiven by the number 1  2  3  …  (N – 1), called the factorialof(N – 1)and written as (N – 1)! for short.

  15. NUMBER OF HAMILTON CIRCUITS IN KN ■KNhas N(N – 1)/2edges. ■Of all graphs with N vertices and no multiple edges or loops, KN has the most edges. The table on the next slide shows the number of Hamilton circuits in complete graphs with upto N = 20vertices.

  16. Number of Hamilton Circuits The main point of the table is to convince you that as we increase the number of vertices, the number of Hamilton circuits in the completegraph goes through the roof. Even a relatively small graph such as has K8 morethan five thousand Hamilton circuits. Double the number of vertices to K16andthe number of Hamilton circuits exceeds 1.3 trillion. Double the number of vertices again to K32and the number of Hamilton circuits – about eight billion trillion trillion – is so large that it defies ordinary human comprehension.

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