Euler and Hamiltonian Circuits

Euler and Hamiltonian Circuits As I type this sentence millions of students all over the country are in their math class either a) struggling to open their eyelids or b) tapping their fingers due to boredom and impatience. They have all failed to understand how the topic would later come of use. Although mathematics may seem to be “unnecessary” it teaches our brains to strategize, and think differently through the use of trial and error and problem solving. Most individuals consider mathematics to be a dreadful topic, and can never really comprehend how it can be beneficial in our lives on a daily basis.
Most of the time, they may seem to be correct. However, they are not. Most of the time, we are using its strategies without even acknowledging it. We use it for almost anything we do: currency, measurement, time, etc. Two examples of math we use on a regular basis are Euler and Hamiltonian Circuits. An Euler Circuit is a circuit that reaches each edge of a graph exactly once. (Malkevitch, 8) This theory is named after Leonhard Euler, an outstanding mathematician during the 18th century. Euler had been the first person to study this category of circuits.
In addition, he was the creator of the theory of graphs, or graph theory. One of the many things he had found was that most graphs do not have an Euler circuit at all. Euler had also contributed to the field of mathematics in various ways. He was a very creative individual, establishing more than 500 works throughout his lifetime. Euler had been considered a prodigy because he was working with the most complex mathematical calculations under the very poor conditions he lived in, and proceeded to work with these problems until he had become totally blind. Malkevitch, 9) According to Professor Clark Kimberling, some of the other things Euler had discovered or had named after him in his honor are: e (the calculus number), a,b,c (the side lengths of a triangle), f(x) (for functional value), R and r (the circumradius and inradius of a triangle), sin x and cos x (values of sine and cosine functions), i (for the square root of -1), capital sigma (summation), and, lastly, capital delta (finite difference). In 1736, Euler had come up with the idea of a graph when he held the ability to solve a problem in “recreational mathematics”.

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He had proven that it was not possible to go to a route visiting the seven bridges of the German town of Konigsberg only, and not more than, once. A Hamiltonian Circuit is a tour that begins at a vertex of a graph and visits each vertex exactly once, and then returns to where it had originated. (Malkevitch, 35) This theory is named after Sir William Rowan Hamilton, an Irish mathematician and astronomer, who lived from 1805 to 1865. Much like Euler, Hamilton was considered a prodigy except as a child.
He had the ability to read four dialects (English, Hebrew, Greek, and Latin) by the time he reached the age of four. Additionally, he enjoyed writing poetry and was able to maintain close friendships with other well-known scholars such as Wordsworth and Coleridge. When he was just twenty-three years old, he became a Professor of Astronomy in Trinity College in Dublin, Ireland. (Bowen Larry) He received the privilege of having Hamiltonian Circuits named after him because he was one of the first to study it.
However, the first to discover this concept was Thomas Kirkman, a British minister with a liking for mathematics, who lived from 1806 to 1895. These two concepts are significant to the field of mathematics because they are the appropriate devices for analyzing problems where it is necessary to visit each vertex or edge only once. People often associate Hamiltonian Circuits with Euler Circuits because they both prohibit reuse. Euler Circuits prohibits the reuse of edges whereas Hamiltonian Circuits do not allow the reuse of vertices.
Both Euler and Hamiltonian circuits are extremely beneficial in our daily lives because they are classified under problems known as “routing problems”. According to Professor Larry Bowen, “routing problems” are problems whose solution attempts the most efficient way(s) of routing things among different destinations. These problems may appear in various areas such as: transportation, communications, and the delivery of services. Without our realization, we use Euler problems on a day-to-day basis specifically in management science.
Anytime it is necessary for services to go through streets or roads the Euler Circuit theory can be more resourceful. To show more specific problems, Euler circuits can help an individual while they are salting icy roads, plowing snow, mowing grass along highways, collecting garbage, collecting debris or leaves from urban curbs, inspecting railroad tracks for flaws, or reading electric meters at private houses. (Malkevitch, 19) We use them to indicate where the arrows on the streets should be placed, and in what direction they should go in.
In addition, we use them to construct the schedules for when parking is forbidden. Because parking-times are massive factors in street sweeping, it is essential find a circuit that visits streets when they are free of cars. Also, we use Euler circuits to distribute territory into multiple routes for street sweepers, parking officers, and sanitation workers. The objective is to find easy routes for them to travel by as well as taking traffic direction, number of lanes, parking-time restrictions, and divided routes into consideration.
All of these examples are possible through management science, a field that most people would want to attain as a career in one day. (Malkevitch, Joseph) Just like Euler circuits problems, we essentially use Hamiltonian circuit problems on a daily basis as well specifically through the use of business efficiency. Hamiltonian circuits have many applications. The deliveries of mail and packages, or water meter inspections are done with the use of Hamiltonian circuit problems because it is necessary that they meet each vertex within a graph.
Some examples of its regular uses are for inspecting traffic signals, for delivering mail to drop-off boxes, or for delivering Meals on Wheels to the elderly. (Malkevitch, 35) In addition, an individual can use a Hamiltonian Circuit when they plan on going on a vacation. They would construct a graph of all the places they would like to visit and figure the distances between all of the places, and essentially they would be able to figure out which is the best possible route to take as well as with the shortest distance.
As shown in the examples above, Euler and Hamiltonian circuits have made advancements to their field because they give individuals assistance in transportation, communications, and delivery of services. They give all of the following areas the ability to come up with routes in a well-organized, efficient fashion. On a personal level, these types of circuits have interested me not only because of the history of the individuals who created them, but also because of what these circuits are used for on a daily basis.
At first, when I had learned about the topic in class I was only focused on solely the arithmetic and formulas of the problems. Later, while researching into their uses, I was then aware of how they can be used in real-life situations. With a background of this information, if I ever wish to pursue a career in management science or business efficiency, I will have a general idea of how it works. Overall, these circuits can teach an individual how mathematics comes into play in real-life situations. Works Cited 1. Bowen, Larry, Dr. “Quick Summary. ” Quick Summary. University of Alabama, n. . Web. 28 Nov. 2012. . 2. Malkevitch, Joseph. “Chapter 2: Business Efficiency. ” For All Practical Purposes: Mathematical Literacy in Today’s World. 9th ed. New York: W. H. Freeman and, n. d. 35-57. Print. 3. Malkevitch, Joseph. “Chapter 1: Urban Services. ” For All Practical Purposes: Mathematical Literacy in Today’s World. 9th ed. New York: W. H. Freeman and, n. d. 5-21. Print. 4. Kimberling, Clark. “Leonhard Euler. ” LEONHARD EULER. University of Evansville, n. d. Web. 29 Nov. 2012. . 5. Bowen, Larry, Dr. “Introduction. ” Introduction. University of Alabama, n. d. Web. 29 Nov. 2012. .

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