Over the years, the concept of Hamiltonian

circuits has been used in different sectors

especially when deciding the shortest route in order to increase

efficiency. Hamiltonian Circuit is a

consecutive directed edge that uses all the vertex of a graph once. Hamilton’s circuit also referred to as

Hamiltonian Cycle. It is a closed loop representation that comes in contact

with the node once. Hamiltonian Circuits is

renamed after the inventor, Sir William

Rowan Hamilton who also devised Hamilton’s puzzle modernly known as from a

simple game known as Icosian.It entails

locating the Hamiltonian circuit in the graphs apex. It is important to note

that a graph is Hamilton only when there a Hamilton cycle is contained.

Mathematically a Hamiltonian Path is either a directed or undirected graph that

comes in contact with the vertex once. The only exception is that the vertex is

provided at the beginning and termination art that is checked two times in

graphical theory and representation the Hamiltonians path is a series of

vertices and edges whereby every vertex is included once.

Some properties of Hamiltonian circuits

include conversion from a Hamiltonian path to a Hamiltonian cycle can be

achieved via removal of one of the edges (Bode, Anika, and Arnfried).

Adjustments can be made to the Hamiltonian path and the possibility of

extension to become a Hamiltonian cycle only when the endpoints are adjacent.

Any connection with more than two vertices is Hamiltonian only when there is a

strong connection. There are assumptions that the cycles are similar apart from

the beginning point and thus are counted together. There is a broad figure of

the various Hamiltonian Cycles contained in a comprehensive purposeful graph of

n- apexes that mean (n-1)! /2 whereas in

a far-reaching concentrating graph of n-vertices that is simply (n-1)!

Real world

application of Hamiltonian circuits

People use the Hamiltonian circuits in so

many 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 everyday

problems. If the vertices are to represent stationary points and the edges to

stand in for connection points such as roads between the points which in real

life can represent institutions such as Mailboxes,

shops, and malls. The important aspect is

to move between points thus visit every stationary point.

According to Breuckmann, and Barbara, instances

which require an individual to check in

all locations such, as traveling

salesperson that has intent on finding

the optimal Hamiltonian’s path. In the resultant problem, the edges of the graph have

a weight that is associated with

it in form of numbers. Here the salesperson needs to undertake the Hamiltonian

path with the least weight in order to cut on travel

costs and time. An advantage enjoyed is that once someone has come up with an

algorithm to find a Hamiltonian cycle he is now left with the chance to select

the best cycle to solve the traveling salesman problem

There are

several methods for solving the Travelling Sales Person problem that use

algorithms to go about it. Some of the examples of these approaches are the

brute force algorithm, cheapest link algorithm, nearest

neighbor algorithm, and repetitive

nearest neighbor algorithm.

The brute

force algorithm

A number of steps are followed to ensure

that this algorithm works. Ideally, one

starts with identifying all possible Hamilton circuits without indicating the

exact reversals, secondly find the weight of them and finally select the path

with the smallest weight. The main advantage is that when given ample time and

space it works always without fail (Audibert, 835-865). However, on the downside, it

can only be used for small representations

Cheapest

link algorithm

In order to efficiently utilize this method, an individual has to select the edge

with the smallest weight that is often substituted for the cheapest. An

individual has to follow this process continuously unless there is a collision

with 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. The

resulting solution is the Hamilton Circuit

Nearest

neighbor algorithm

To

understand this technique it is important to visualize and map out the problem

equating 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 the

neighborhood 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 random

selection is conducted. This process continues until one has traveled all the cities. Finally one travels

back to the vertex (one’s home). The resulting path is the Hamilton circuit.

Repetitive

nearest neighbor algorithm

This algorithm uses similar concept to the nearest neighbor algorithm (Audibert,

835-865). Here one chooses the best solution (the one that has the smallest

weight). Should there be a need for an individual to rewrites this solution

starting with a particular vertex that is considered home. Hamilton circuit can also be applied when finding a Knight’s

tour (Gould,18). This is a challenge to get the knight to move to every space

on board of chess only once. This is equitable to locating a Hamiltonian path

in the graph and hereby describing all the possible moves of the knight on the

chess board. This illustration is relevant in that the graph of the Knight’s

moves has a Hamiltonian path.

Other illustrations for the Hamilton

circuit include garbage truck, bus

terminus, tyre pressure readings.

Hamiltonian circuits help us solve the routing programs. These are problems

that the solution attempt the ideal way

for routing and directing things among different locations. They are broadly

instances in transport, communication and service delivery, water meter

inspections, mail delivery. This is necessary since they meet at the vertex

within a group.

Additional instances where the Hamiltonian

circuit is regularly used include traffic signals inspection and delivery of

meals to the elderly. When planning for a vacation whereby the areas to be

visited are graphed, a decision is made on the best route to take and also one

that has the shortest distance. The Hamiltonian circuit helps in the planning of routes activities, saves collision

that would occur due to route mismanagement and simplification of tasks that

they appear seamless (Gould, 23). They also help in management and control of

traffic without the need for supervision.

In mathematics and related fields, the Hamiltonian path also referred to

as the traceable path is that defined path that is either directed or

undirected graph that visits the vertex singularly (Gould,33). To identify

whether a graph shall exhibit a Hamiltonians path there are ways devised to

achieve that. For the basic graphs ideally,

that can be done by hand however for complex graphs a direct approach is not

appropriate as it will not provide a solution in record time. The existence of

Hamiltonians circuit mathematicians would always be forced to exhaust any

possible clue, link or path in order to prove it. This is tiresome and could be

a lifelong experience. Large and

difficult graphs could be realized.

Conclusion

The Hamilton circuit helps mathematicians

to optimally find solutions at a faster rate whether for simple or complicated

graphs. The Icosian game that Sir William Hamilton introduced later came to be

known as the Hamilton Circuit problem. The main objective of this game was to

ensure that all nodes and vertices were visited ideally at a single passing

before returning to the start node. Graphically speaking a Hamiltonians circuit

is referred to as a basic simple complete cycle that contains every vertex of

the graph with the only exception being made to the visited to complete the

cycle. A Hamilton graph is said to be so if it contains a Hamilton circuit. The

Hamilton circuit helps mathematicians to optimally find solutions at a faster

rate whether for simple or complicated graphs. A search execution procedure is

used to verify whether Hamilton circuits exist in a given graph and identifies

all of them.