Writing Project Topics Here is a list of places in the text to look for writing project material.This is not a complete list of possibilities, but could be of use to anyone who just hasn't thought of...

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Writing Project Topics Here is a list of places in the text to look for writing project material.This is not a complete list of possibilities, but could be of use to anyone who just hasn't thought of anything yet.? Countability and uncountability (2.5)Just about anything in chapter 3, such as...What is complexity?What are P and NP? EtcCompare various sorting algorithmsCryptography (4.6)5.4 is excellent computer science peopleEvery section of chapter 8 has good stuff.8.2 is a computational-type problem with wide practical application8.3 is great for coputer science types8.


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Writing Project Topics Here is a list of places in the text to look for writing project material. This is not a complete list of possibilities, but could be of use to anyone who just hasn't thought of anything yet. ? Countability and uncountability (2.5) Just about anything in chapter 3, such as... What is complexity? What are P and NP? Etc Compare various sorting algorithms Cryptography (4.6) 5.4 is excellent computer science people Every section of chapter 8 has good stuff. 8.2 is a computational-type problem with wide practical application 8.3 is great for coputer science types 8.4 is deep magic for the mathematicians in the room There's so much juicy stuff in chapter 10 that we don't have time to do justice … Hamiltonian circuits and paths … this is an open research area; discuss some of the known results about which graphs do and don't have Hamiltonian circuits… algorithms … etc Graph coloring? The four-color problem. Discuss a few applications of trees from 11.2, and find and solve some associated problems Chapter 11.4/11.5, especially a comparison of various algorithms to find spanning trees of various kinds Chapter 12, Boolean algebra, but an interesting backspin on the proposition logic that began the course. 12.3 and 12.4 in particular seem like good places to look ? Writing Project Grading Rubric (Revised) Writing Project Rubric Project Attributes Properties of an Excellent Paper The paper discusses some topic in mathematics related to the material of the course; if it is not obvious, the paper makes clear what the connection is to the course. The amount of material presented is suitable for giving a presentation of about 20 minutes (longer for a group of three). I can tell from reading the paper that you have learned something. I can tell from reading the paper that you have actually done some creative mathematics.  (This does not require you to do original research, of course; this could be satisfied by developing examples not found in your...



Answered Same DayDec 23, 2021

Answer To: Writing Project Topics Here is a list of places in the text to look for writing project...

Robert answered on Dec 23 2021
133 Votes
Graph Coloring
Let G = (V, E) represent a graph with vertex set V and edge set E formed by all pairs of incompatible
Elements.
The problem says; suppose that V contains certain pairs of elements that are incompatible. Our aim is to

find a partition of V into a minimal number of subsets of mutually compatible elements.
When we talk about partitioning of V into k subsets, it is equivalent to coloring the vertices of G with k
colors.
A (vertex) colouring of a graph G is a mapping C : V(G) → S.
The elements of S are called colours; the vertices of one colour form a colour class. The colour classes
divide V into independent subsets ; subset of pair wise non-adjacent vertices.
If |S| = k, we say that c is a k-colouring (often we use S = {1, . . . , k}).
A colouring is proper when no two vertices share the same colour.
A graph is k-colourable if it has a proper k-colouring. The chromatic number X(G) is the least k such that
G is k-colourable. Obviously, X(G) exists as assigning distinct colours to vertices yields a proper
|V(G)|-colouring.
An optimal colouring of G is a X(G)-colouring. A graph G is k-chromatic if X(G) = k.
The problem of determining X(G), the chromatic number, is an NP - complete.
An NP - complete problem has no polynomial time algorithm.
A 3-colour example of a vertex colouring is shown below :
Chromatic Polynomial :
A chromatic polynomial counts the number of ways a graph can be colored, using no more than the
number of colours given. That is, it counts the number of its proper vertex colourings.
We have used X(G) to denote a chromatic number.
A chromatic polynomial can be represented as P(G, k) where G represents the graph, and k denotes the
number of colours. Thus, for a given G, the function is a polynomial in k.
for example :
P(G, k) = k(k-1)(k-2)(k-3)
=> P(G, 4) = 24
The chromatic polynomial P(G, k) and the chromatic number X(G) are related. the chromatic number is
the smallest positive integer that is not a root of the chromatic polynomial,
X(G) = min { k : P(G, k) > 0 }
Common Properties of P(G, k) :
Evaluating the chromatic polynomial in P(G, k) yields the number of k-colorings of G for k = 0, 1, 2, .. , n
for all n є N.
In the graph G; n be the number of vertices, m be the edges and s be the components G1, G2, .., Gs. Then
:
 The...
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