The chromatic polynomial is a concept in algebraic graph theory that was first studied by George David Birkhoff in 1912. As a function of the number of colours, it evaluates all possible distinct vertex colourings of a given graph. Using the edge deletion and contraction method, the chromatic polynomial can be expressed as a sum of simpler subgraphs. Hence, it can be evaluated recursively. Many relevant properties can also be proven inductively. George originally studied the chromatic polynomial in an attempt to prove the Four Colour Problem, which states that any map can be coloured using at most 4 colours such that no two adjacent regions have the same colour. In the context of graph theory, any map can be seen as a planar graph. George wanted to show that 4 is not a chromatic number (root) for planar graphs in general. Presently, the roots and coefficients of the chromatic polynomial remain as important subjects of study in modern day graph theory.
Chapter 1 will formally introduce the vertex colouring problem. Some of the most important results in relevance to vertex colouring will be discussed.
Chapter 2 is a rigorous study on the chromatic polynomial in the context of algebraic graph theory.
Chapter 3 and 4 are based on personal research. Some interpretations of the coefficients and conditions on the roots of the chromatic polynomial will be discussed.
Chapter 1
Vertex Colouring Problem
The structure of a graph Γ is defined by its vertices and their pairwise relations specified by the edges. A vertex colouring is an assignment of colours to the vertices of Γ and therefore defines a partition of its vertex set.
Definition 1.1. Let Γ = (V, E) be a finite graph, where V is the set of vertices and E the set of edges. A vert...
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...(c) = M4(e) = M5(b) = M5(d).
M4 : V → C M5 : V → C
Figure 1.4: the disconnected bipartite graph Γ3 is not uniquely colourable
If Γ is uniquely colourable, then there exists one and only one feasible partition of V into χ(Γ) = r independent sets. Hence, if M is an r-colouring of Γ, then every vertex in any colour class Vi of M must have at least one edge connecting to any other colour classes. Thus,
│E│ ≥ ((r-1)│V│)/2 .
In fact, a better bound can be found, which leads to the next proposition.
Proposition 1.7. Let Γ = (V, E) be a finite graph and χ(Γ) = r. If Γ is uniquely colourable, then
│E│ ≥ (r-1)│V│ - (r(r-1))/2 .
Proof. Let P = {V1, V2, …, Vr} be the unique partition of V into r independent subsets. Note that every subgraph Γ’(Vi ) induced by any two colour classes Vi and Vj must be
The context in which Marc worked suggests he employed these correspondences programmatically, Marc wrote: "Every color must say clearly 'who and what it is, and must, moreover, be related to a clear form."
Master's Theses. Paper 3505. http://scholarworks.sjsu.edu/etd_theses/3505 Version, H. B. & Co., Ltd. (1984) The 'Standard' of the Keith (Ed) Danby.
When it comes to Gatsby, these colors represent him in many ways. Red is by far the most used color in this book. It is used so much because it has so many meanings. The color red is interchangeable with the colors crimson, ruby, blood, and rose. Red comes in various shades too, but with each different shade comes a different meaning.
reference should consult with Colin Stanley’s book, and its recent supplement. I have also appended a list of his shorter fiction.
The Great Gatsby is full of symbolism, colors, for example. Throughout the book the author uses them to represent different themes of the novel. Some of these colors are white, yellow, grey, green, pink, red and blue. However, I picked white and green for my commentary because I think these colors have a special meaning different from the others. White is mainly used to describe the character’s innocence, fakeness, and corruption. While green represents Gatsby’s hopes, ambitions, and dreams. In addition, sometimes green symbolizes the jealousy of certain characters.
If one were to trace the color red through the book, it would be almost impossible to give it one decisive meaning- and that is the point entirely. The color red appears to symbolize not
Throughout history race has been a pivotal part in decision making. Since the first interaction of populations from different geographical regions, with physical differences, the assumption has been that these differences were fundamental. These physical differences, based mainly on the color of skin, have been classified as race. Over time physicians have tried to pinpoint certain characteristics that are more prominent in one race than another, in order to prove race and that there are subspecies of humans. One of the first examples of this is Johann Blumenbach work “On the Natural Variety of Mankind”. In his work Blumenbach introduced five categories of race; caucasian. mongolian, malayan ethiopian, and american. These classifications revolved around not only skin color, but also physical morphology. Blumenbach supported this by using a collection of physical evidence, like the structure of the cranium, to support his thesis. This marked a shift in thinking of race in terms of geography to physical appearance. Published in 1775, Blumenbach’s dissertation started the scientific approach to finding
Pratt utilized blues, greens, and browns as the background, creating a calming analogous color scheme. The placement of the beautiful red-orange design throughout the middle of the painting gives the image a sharp contrast against the blue-greens found behind it, creating a complementary color scheme. All the colors in Garden tell a beautiful story. The hues found in the background give a sense of walking through a calm, serene garden. The bright red-orange color strewn through the middle evoke the feeling of love, but also a feeling of anger or frustration, as it almost disrupts the calming colors that it has placed on top of. Altogether, a multitude of emotions can be felt as one moves through the many hues in the
In books Authors us colors sometimes to represent something. Colors have a way bigger meaning as F. Scott Fitzgerald uses them. The colors I choose were green, white, gold, and grey.
Landesman defends a view called color skepticism, that nothing has any color, neither bodies nor appearances. He came to the conclusion that colors do not exist. In making the case for his "color skepticism," Landesman discusses and rejects historically influential
Since hundred years ago, when people started to make maps to show distinct regions, such as states or countries, the four color theorem has been well known among many mapmakers. Because a mapmaker who can plan very well, will only need four colors to color the map that he makes. The basic rule of coloring a map is that if two regions are next to each other, the mapmaker has to use two different colors to color the adjacent regions. The reason is because when two regions share one boundary can never be the same color. Another basic rule of coloring a map is that if two regions share only one point, then they do not necessary have to be colored differently. Many evidence showed that coloring a map required at least four colors but no more than five. Then mathematicians started to asked questions, such as “ Is it true that using only four colors are enough? Is there any exception that one has to color a map that requires more than four colors? Or is it has to do with a special sequence of arrangement that involved with different regions in order to make the theorem true?” However, the first mathmatician who asked these questions is a man named Francis Guthrie. He was the first one who posed the four color problem in1852.
...y) that pools all of these colors together. The colors could represent the varying aspects and cultures contained within the Human Race. The choices presented both in the title and in the story itself create a vast labyrinth, much in the way Borges conceives the workings of an elaborate universe.
Hochschild, Jennifer L. “The Skin Color Paradox and the American Racial Order.” Scholar. Harvard.edu, December.2007.
In a world without color everything would be black and white. Living a life without color would be dull and boring. “ The life where nothing was ever unexpected. Or inconvenient. Or unusual. The life without color, pain, or past.” To live a life without color would be like drawing a beautiful picture without coloring it in. The world is the picture and without color it would feel as if something was missing. I would give up a life of sameness, of not making choices to be able to see color. I feel as if color is an important way to express emotions. Colors stand for feelings, for example yellow means happy.
Colors are important when one is describing a feeling, character, and setting. They can be used to assume a person 's personality when the author takes the time to describe the color of someone’s clothes. An author can also create certain feelings about the day or night in the scene by the colors they use to describe it. Symbolism is the heart of literature and color is one type of symbolism that is used. In