Over the years, the concept of Hamiltoniancircuits has been used in different sectorsespecially when deciding the shortest route in order to increaseefficiency. Hamiltonian Circuit is aconsecutive directed edge that uses all the vertex of a graph once. Hamilton’s circuit also referred to asHamiltonian Cycle. It is a closed loop representation that comes in contactwith the node once. Hamiltonian Circuits isrenamed after the inventor, Sir WilliamRowan Hamilton who also devised Hamilton’s puzzle modernly known as from asimple game known as Icosian.It entailslocating the Hamiltonian circuit in the graphs apex.
It is important to notethat a graph is Hamilton only when there a Hamilton cycle is contained.Mathematically a Hamiltonian Path is either a directed or undirected graph thatcomes in contact with the vertex once. The only exception is that the vertex isprovided at the beginning and termination art that is checked two times ingraphical theory and representation the Hamiltonians path is a series ofvertices and edges whereby every vertex is included once.
Some properties of Hamiltonian circuitsinclude conversion from a Hamiltonian path to a Hamiltonian cycle can beachieved via removal of one of the edges (Bode, Anika, and Arnfried).Adjustments can be made to the Hamiltonian path and the possibility ofextension to become a Hamiltonian cycle only when the endpoints are adjacent.Any connection with more than two vertices is Hamiltonian only when there is astrong connection.
There are assumptions that the cycles are similar apart fromthe beginning point and thus are counted together. There is a broad figure ofthe various Hamiltonian Cycles contained in a comprehensive purposeful graph ofn- apexes that mean (n-1)! /2 whereas ina far-reaching concentrating graph of n-vertices that is simply (n-1)!Real worldapplication of Hamiltonian circuitsPeople use the Hamiltonian circuits in somany ways without even acknowledging it. Hamiltonian circuits can be applied to currency calculations and conversion,various measurements as well as in time variance calculations(Breuckmann, and Barbara).The Hamiltonian circuits help solve everydayproblems.
If the vertices are to represent stationary points and the edges tostand in for connection points such as roads between the points which in reallife can represent institutions such as Mailboxes,shops, and malls. The important aspect isto move between points thus visit every stationary point.According to Breuckmann, and Barbara, instanceswhich require an individual to check inall locations such, as travelingsalesperson that has intent on findingthe optimal Hamiltonian’s path. In the resultant problem, the edges of the graph havea weight that is associated withit in form of numbers. Here the salesperson needs to undertake the Hamiltonianpath with the least weight in order to cut on travelcosts and time. An advantage enjoyed is that once someone has come up with analgorithm to find a Hamiltonian cycle he is now left with the chance to selectthe best cycle to solve the traveling salesman problemThere areseveral methods for solving the Travelling Sales Person problem that usealgorithms to go about it. Some of the examples of these approaches are thebrute force algorithm, cheapest link algorithm, nearestneighbor algorithm, and repetitivenearest neighbor algorithm.
The bruteforce algorithmA number of steps are followed to ensurethat this algorithm works. Ideally, onestarts with identifying all possible Hamilton circuits without indicating theexact reversals, secondly find the weight of them and finally select the pathwith the smallest weight. The main advantage is that when given ample time andspace it works always without fail (Audibert, 835-865).
However, on the downside, itcan only be used for small representationsCheapestlink algorithmIn order to efficiently utilize this method, an individual has to select the edgewith the smallest weight that is often substituted for the cheapest. Anindividual has to follow this process continuously unless there is a collisionwith the smaller circuit or the three selected edges pop from a single vertex (Audibert,835-865). This process continues until the Hamilton Circuit is Complete. Theresulting solution is the Hamilton CircuitNearestneighbor algorithmTounderstand this technique it is important to visualize and map out the problemequating it to the neighborhood (Audibert, 835-865).
We commence at a vertex,secondly travel to the vertex which we could equate it to the part of theneighborhood you haven’t yet visited whose path contains the smallest weight(we can represent this as the closest city) but should there be a tie a randomselection is conducted. This process continues until one has traveled all the cities. Finally one travelsback to the vertex (one’s home).
The resulting path is the Hamilton circuit.Repetitivenearest neighbor algorithmThis algorithm uses similar concept to the nearest neighbor algorithm (Audibert,835-865). Here one chooses the best solution (the one that has the smallestweight). Should there be a need for an individual to rewrites this solutionstarting with a particular vertex that is considered home. Hamilton circuit can also be applied when finding a Knight’stour (Gould,18).
This is a challenge to get the knight to move to every spaceon board of chess only once. This is equitable to locating a Hamiltonian pathin the graph and hereby describing all the possible moves of the knight on thechess board. This illustration is relevant in that the graph of the Knight’smoves has a Hamiltonian path.Other illustrations for the Hamiltoncircuit include garbage truck, busterminus, tyre pressure readings.Hamiltonian circuits help us solve the routing programs. These are problemsthat the solution attempt the ideal wayfor routing and directing things among different locations. They are broadlyinstances in transport, communication and service delivery, water meterinspections, mail delivery. This is necessary since they meet at the vertexwithin a group.
Additional instances where the Hamiltoniancircuit is regularly used include traffic signals inspection and delivery ofmeals to the elderly. When planning for a vacation whereby the areas to bevisited are graphed, a decision is made on the best route to take and also onethat has the shortest distance. The Hamiltonian circuit helps in the planning of routes activities, saves collisionthat would occur due to route mismanagement and simplification of tasks thatthey appear seamless (Gould, 23). They also help in management and control oftraffic without the need for supervision.
In mathematics and related fields, the Hamiltonian path also referred toas the traceable path is that defined path that is either directed orundirected graph that visits the vertex singularly (Gould,33). To identifywhether a graph shall exhibit a Hamiltonians path there are ways devised toachieve that. For the basic graphs ideally,that can be done by hand however for complex graphs a direct approach is notappropriate as it will not provide a solution in record time. The existence ofHamiltonians circuit mathematicians would always be forced to exhaust anypossible clue, link or path in order to prove it. This is tiresome and could bea lifelong experience. Large anddifficult graphs could be realized. Conclusion The Hamilton circuit helps mathematiciansto optimally find solutions at a faster rate whether for simple or complicatedgraphs.
The Icosian game that Sir William Hamilton introduced later came to beknown as the Hamilton Circuit problem. The main objective of this game was toensure that all nodes and vertices were visited ideally at a single passingbefore returning to the start node. Graphically speaking a Hamiltonians circuitis referred to as a basic simple complete cycle that contains every vertex ofthe graph with the only exception being made to the visited to complete thecycle. A Hamilton graph is said to be so if it contains a Hamilton circuit. TheHamilton circuit helps mathematicians to optimally find solutions at a fasterrate whether for simple or complicated graphs.
A search execution procedure isused to verify whether Hamilton circuits exist in a given graph and identifiesall of them.