# Graph Theory

**Topics:**Graph theory, Planar graph, Graph coloring

**Pages:**18 (2022 words)

**Published:**November 8, 2012

Sharathkumar.A,

Final year, Dept of CSE,

Anna University, Villupuram

Email: kingsharath92@gmail.com

Ph. No: 9789045956

Abstract

Graph theory is becoming increasingly significant as it is applied to other areas of mathematics, science and technology. It is being actively used in fields as varied as biochemistry (genomics), electrical engineering (communication networks and coding theory), computer science (algorithms and computation) and operations research (scheduling). The powerful combinatorial methods found in graph theory have also been used to prove fundamental results in other areas of pure mathematics. We discuss about computer network security (worm propagation) using minimum vertex covers in graphs. We also show how to apply edge coloring and matching in graphs for scheduling (the timetabling problem) and vertex coloring in graphs for map coloring and the assignment of frequencies in GSM mobile phone networks. Finally, we revisit the classical problem of finding re-entrant knight’s tours on a chessboard using Hamiltonian circuits in graphs.

Introduction

Graph theory is rapidly moving into the mainstream of mathematics mainl y because of its applications in diverse fields which include biochemistry (genomics), electrical engineering (communications networks and coding theory), computer science (algorithms and computations) and operations research (scheduling). The wide scope of these and other applications has been well-documented cf. [5] [19]. The powerful combinatorial methods found in graph theory have also been used to prove significant and well-known results in a variety of areas in mathematics itself. An application of matching in graph theory shows that there is a common set of left and right coset representatives of a subgroup in a finite group. This result played an important role in Dharwadker’s 2000 proof of the four-color theorem [8] [18]. The existence of matchings in certain infinite bipartite graphs played an important role in Laczkovich’s affirmative answer to Tarski’s 1925 problem of whether a circle is piecewise congruent to a square. The proof of the existence of a subset of the real numbers R that is non-measurable in the Lebesgue sense is due to Thomas [21]. Surprisingly, this theorem can be proved using only discrete mathematics (bipartite graphs). There are many such examples of applications of graph theory to other parts of mathematics, but they remain scattered in the literature [3][16]. In this paper, we present a few selected applications of graph theory to other parts of mathematics and to various other fields in general.

Applications

Computer Network Security

A team of computer scientists led by Eric Filiol [11] at the Virology and Cryptology Lab, ESAT, and the French Navy, ESCANSIC, have recently used the vertex cover algorithm [6] to simulate the propagation of stealth worms on large computer networks and design optimal strategies for protecting the network against such virus attacks in real-time.

Figure 5.1. The set {2, 4, 5} is a minimum vertex cover in this computer network The simulation was carried out on a large internet-like virtual network and showed that that the combinatorial topology of routing may have a huge impact on the worm propagation and thus some servers play a more essential and significant role than others. The real-time capability to identify them is essential to greatly hinder worm propagation. The idea is to find a minimum vertex cover in the graph whose vertices are the routing servers and whose edges are the (possibly dynamic) connections between routing servers. This is an optimal solution for worm propagation and an optimal solution for designing the network defense strategy. Figure 5.1 above shows a simple computer network and a corresponding minimum vertex cover {2, 4, 5}.

TimeTabling Problem

In a college there are m professors x1, x2, …, xm and n subjects y1, y2, …, yn to be taught. Given...

References: Ashay Dharwadker, The Vertex Coloring Algorithm,

2006, http://www.dharwadker.org/vertex_coloring

Ashay Dharwadker, A New Algorithm for finding Hamiltonian Circuits,

2004, http://www.dharwadker.org/hamilton

[10] L. Euler, Solution d 'une question curieuse qui ne paroit soumise a aucune analyse, Mémoires de

l 'Académie Royale des Sciences et Belles Lettres de Berlin, Année 1759 15, 310-337, 1766.

[12] K. Heinrich and P. Horak, Euler’s theorem, Am. Math. Monthly, Vol. 101 (1994) 260.

Leipzing (1936), reprinted by Chelsea, New York (1950).

[17] H. J. R. Murray, A History of Chess, Oxford University Press, 1913.

[18] Shariefuddin Pirzada and Ashay Dharwadker, Graph Theory, Orient Longman and Universities

Press of India, 2007.

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