# 数学代写|离散数学作业代写discrete mathematics代考|MATH-UA120

## 数学代写|离散数学作业代写discrete mathematics代考|Graph Colouring and Four-Colour Problem

It is very common for maps to be coloured in such a way that neighbouring states or countries are coloured differently. This allows different states or countries to be easily distinguished as well as the borders between them. The question naturally arises as to how many colours are needed (or determining the least number of colours needed) to colour the entire map, as it might be expected that a large number of colours would be needed to colour a large complicated map.

However, it may come as a surprise that in fact very few colours are required to colour any map. A former student of the British logician, Augustus De Morgan, had noticed this in the mid-1800s, and he proposed the conjecture of the four-colour theorem. There were various attempts to prove that 4 colours were sufficient from the mid-1800s onwards, and it remaincd a famous unsolved problem in mathematics until the late twentieth century.

Kempe gave an erroneous proof of the four-colour problem in 1879, but his attempt led to the proof that five colours are sufficient (which was proved by Heawod in the late $1800 \mathrm{~s})$. Appel and Haken of the University of Illinois finally provided the proof that 4 colours are sufficient in the mid-1970s (using over $1000 \mathrm{~h}$ of computer time in their proof).

Each map in the plane can be represented by a graph, with each region of the graph represented by a vertex. Edges connect two vertices if the regions have a common border. The colouring of a graph is the assignment of a colour to each vertex of the graph so that no two adjacent vertices in this graph have the same colour.
Definition
Let $\mathrm{G}=(\mathrm{V}, \mathrm{E})$ be a graph and let $\mathrm{C}$ be a finite set called the colours. Then, a colouring of $\mathrm{G}$ is a mapping $\kappa: \mathrm{V} \rightarrow \mathrm{C}$ such that if $u v \in \mathrm{E}$ then $\kappa(u) \neq \kappa(v)$.
That is, the colouring of a simple graph is the assignment of a colour to each vertex of the graph such that if two vertices are adjacent then they are assigned a different colour. The chromatic number of a graph is the least number of colours needed for a colouring of the graph. It is denoted by $\chi(G)$.

## 数学代写|离散数学作业代写discrete mathematics代考|Cryptography

Cryptography was originally employed to protect communication of private information between individuals. Today, it consists of mathematical techniques that provide secrecy in the transmission of messages between computers, and its objective is to solve security problems such as privacy and authentication over a communications channel.

It involves enciphering and deciphering messages, and it employs theoretical results from number theory to convert the original message (or plaintext) into cipher text that is then transmitted over a secure channel to the intended recipient. The cipher text is meaningless to anyone other than the intended recipient, and the recipient uses a key to decrypt the received cipher text and to read the original message.

The origin of the word “cryptography” is from the Grcek ‘kryptos’ mcaning hidden, and ‘graphein’ meaning to write. The field of cryptography is concerned with techniques by which information may be concealed in cipher texts and made unintelligible to all but the intended recipient. This ensures the privacy of the information sent, as any information intercepted will be meaningless to anyone other than the recipient.

Julius Caesar developed one of the earliest ciphers on his military campaigns in Gaul. His objective was to communicate important messages safely to his generals. His solution is one of the simplest and widely known encryption techniques, and it involves the substitution of each letter in the plaintext (i.e., the original message) by a letter a fixed number of positions down in the alphabet. The Caesar cipher involves a shift of 3 positions and this leads to the letter $\mathrm{B}$ being replaced by E, the letter $\mathrm{C}$ by $\mathrm{F}$, and so on.

The Caesar cipher (Fig. 10.1) is easily broken, as the frequency distribution of letters may be employed to determine the mapping. However, the Gaulish tribes who were mainly illiterate, and it is likely that the cipher provided good security.

# 离散数学代写

## 数学代写|离散数学作业代写discrete mathematics代考|Cryptography

. crypgraphy

“密码学”一词的起源来自希腊语“kryptos”，意为隐藏，“graphein”意为书写。密码学研究的是一种技术，通过这种技术，信息可以隐藏在密文中，除了预期的接受者之外，所有人都无法理解。这确保了所发送信息的隐私性，因为任何被截获的信息对于除收件人以外的任何人都是无意义的

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