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##### Chromatic Number Calculation

1 Introduction to Chromatic Number
This project will investigate the main features of coloring with a network graph thereby further knowledge and awareness about their uses and applications.
Two types coloring scheme will be introduced, their calculation paths and some correlated rules will be considered in order to explore their performances. The primary intention of this project is to investigate the algorithmic method for the two different types of coloring scheme. These will be considered and developed through the use of the Euler Rule and other correlated graph theories.

Contents
1 Introduction to Chromatic Number 2
2 De nitions of chromatic number 2
2.1 Vertex Coloring . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.2 Edge Coloring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3 The importance of Proofs in Mathematics 3
3.1 Fermat Little Theorem. . . . . . . . . . . . . . . . . . . 3
3.2 Chromatic Number. . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.2.1 Vertex-chromatic Number . . . . . . . . . . . . . . . . . . 4
3.2.2 Edge-chromatic Number. . . . . . . . . . . . . . . . . . . 6
4 two example of the real world applications 7
4.1 Example 1. . . . . . . . . . . . . . . .  . . . . . . . 8
4.2 Example 2. . . . . . . . . . . . . . . . . .. . . . . . . 9

References
[1] Reinhard Diestel: Graph Theory.Chromatic Number Calculation S-Verlag Heidelberg,New York
(1997, 2000, 2005)
[2] Ladislav Stacho: New Upper Bounds for the Chromatic Number of a
Graph. Journal of Graph Theory, (2001)
[3] Zhang Xiandi: bulletin of the Institute of Combinatorics and its Applica-
tions. Journal of Graph Theory,On total Coloring of join of Kk
andKn−k
(1993)
[4] Liu Xikui, LiYan: Automation and Logistics. Efficient DNA Algorithms for
Chromatic Number of Graph Problems, (2007)
[5] Borowiecki.P: Information Technology. Algorithmic bounds on the chro-
matic number of a graph (2008)

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