# Graph theory the four coloring theorem essay

None This course provides a detailed examination of the fundamental elements on which computers are based. Topics include number systems and computation, electricity and basic circuits, logic circuits, memory, computer architecture, and operating systems. Some Interesting Issues in Graph Theory. I show three issues in Graph Theory that are interesting and basic. Handshaking Lemma due essentially to Leonhard Euler in If several people shake hands, then what is the total number of hands shaken? Note that it is not the number of handshakes.

When we use some terms of graph theory to think of this question, we can consider a vertex and an edge as a person and a handshake respectively. Since one edge is incident with 2 vertices note that G is simplewe can easily see that 1 handshake consists of 2 people, that is, 2 hands.

This follows that the total number of hands shaken is twice the number of handsake. Unfortunately, we can not tell the exact number of hands shaken because we do not know the number of handshakings. All that we know is that the number is of hands shaken is even.

In terms of graph theory, in any graph the sum of all the vertex-degrees is an even number - in fact, twice the number of edges. Additionally, we can tell that in any graph the number of odd degree vertices is even.

Eulerian graphs Consider a typical problem of asking whether a given diagram can be drawn without lifting one's pencil from the paper and without repeating any lines: Which of the graphs 1, 2, and 3 below can be drawn with a single stroke? The graphs 1 and 2 can be drawn in the desired way, but the graph 3 can not.

Did you see the difference between the way of drawing the graph 1 and that of the graph 2?

[BINGSNIPMIX-3

With regards to the one-stroke drawings of the graph 1, the ending point is the same as the starting point. In contrast, the one-stroke drawings of the graph 2 have a different start and finish. Like the graph 1 above, if a graph has way of getting from one vertex to another that includes every edge exactly once and ends at the initial vertex, then the graph is Eulerian is a Eulerian graph.

Like the graph 2 above, if a graph has ways of getting from one vertex to another that include every edge exactly once and ends at another vertex than the starting one, then the graph is semi-Eulerian is a semi-Eulerian graph. This is equivalent to asking whether the graph below has a Eulerian trail, that is whether the graph is Eulerian.

Did you try it? I guess you could not find any Eulerian trail because Euler proved mathematically that no one can find any one-stroke drawing on that graph. Here is the most famous and simplest theorem Euler about whether a given graph is Eulerian or not.

Hamiltonian graphs While we considered in the "Eulerian graph" section a way of going and returning including every edge of a graph, we consider here a similar problem of going around on a graph including every "vertex" not "edge".

Which of the graphs 1, 2, and 3 below have a way of passing every vertex? The graph 1 and graph 2 has such a way as shown belowbut the graph 3 not.

Similar to the story of Eulerian graph, there is a difference between the way of graph1 and graph 2. That is about the ending points of the paths. With regard to the path of the graph 1, the ending point is the same as the starting point.

In contrast, the path of the graph 2 has a different start and finish. Like the graph 1 above, if a graph has a path that includes every vertex exactly once, while ending at the initial vertex, the graph is Hamiltonian is a Hamiltonian graph.

That path is called a "Hamiltonian cycle". Like the graph 2 above, if a graph has a path that includes every vertex exactly once, but ending at another vertex than the starting one, then the graph is semi-Hamiltonian is a semi-Hamiltonian graph.

Recall that in the previous section of "Eulerian" we saw the very simple and useful theorem about telling whether a graph is Eulerian or not.

Because, unfortunately, little is known in general about Hamiltonian cycle, the finding of such a characterization is one of the unsolved problems of graph theory.Dynamic & original independent Studies.

Unlocking potential with the best learning and research solutions. Pioneer Academics is a U.S.-based organization founded in cooperation between American university professors and international education experts.

An online global research program. Be a Pioneer! Chart your own research journey, mentored by a distinguished professor. Mar 12,  · I've been meaning to write up some tips on interviewing at Google for a good long time now.

I keep putting it off, though, because it's going to make you mad. The four-colour theorem, that every loopless planar graph admits a vertex-colouring with at most four different colours, was proved in by Appel and Haken, using a computer. Topics will be chosen from both pure and applied mathematics and may include algebraic coding theory, cryptology, number theory, mathematical modelling, mathematical logic, complex analysis, topology, dynamical systems, applications to computer science.

The Four Color Theorem. A new proof by Robertson, Sanders, Seymour, and Thomas. Bill Casselman uses postscript to motivate a course in Euclidean geometry. See also his Coxeter group graph paper, and Ed Rosten's Lecture notes from the Clay Math Institute, by Richard Stanley and Federico Ardila, discussing polyomino tilings, coloring.