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There are many ways to color the vertices and edges of graphs, such as, rainbow connection, vertex coloring and dynamic coloring. In this thesis, in Chapter 1 we introduce a new coloring, we consider its relationship with some other colorings and we investigate its computational complexity. In chapter 1, we focus on the proper orientation number. The problem of orienting the edges of a given simple graph so that the maximum indegree of vertices is minimized was introduced in 2004. We show that there is a polynomial time algorithm for determining the proper orientation number of a given 3-regular graph. But it is NP-complete to decide if the proper orientation number of a 4-regular graph is 3... 

Graph coloring is an important concept in graph theory. There have been much developments in this concept recently, vertex coloring, edge coloring and total coloring are studied. But one of the newest kind of graph coloring is [r; s; t] -coloring. This coloring is introduced in 2003 by A. Kemnitz and M. Marangio. In this coloring the difference of any two adjecent vertices, any two adjecent edges, and any adjecent vertex and edge must be at least r, s and t respectively. In this thesis we study [r; s; t] -coloring of graphs and discuss about its results  

In the process of gene assembling, the molecular structure of a DNA chain, can be modeled by a signed graph. After that by means of a composition of three reduction rules: gnr, gpr and gdr, that is called reduction strategy, this graph is reduced to a null graph. If the composition of any ordering of rules in a reduction strategy such as S, is applicable on a signed graph G, then we say that S can be applied in parallel to G and the set S is said to be a parallel step for reduction of that graph. Also we define the least number of parallel steps in reduction of a graph, to be the parallel complexity of that graph, and denote it C(G). In this thesis, a collection of particular signed graphs,... 

In this thesis, we study the application of graph theory in chemistry. Such as, molecular graphs, modeling, algorithms, topologicalindicesandsoon. Themaingoalinthisresearchis collecting problems in chemistry which have mathematical models, specially in graph theory. Also we study the methods applied to them by considering the problems in chemistry with mathematical approach. In particular case, dendrimers and other structures of chemistry have been attributed to some graphs, where by studying their graphical parameters, like connectivity, independent sets, perfect matchings, isomorphism and topological indices and other parameters, we obtain some results in chemistry  

Due to the increasing number of research activities across the world,the academic societies are in the urgent need of appropriate criteria and methods for the evaluation of scientific publications. The term ‘Scientometrics’ has been coined to describe the methodology of aforementioned scientific assessment. Introduction of comprehensive and universal scientific rankings could provide inexperienced researchers with the opportunity to select most qualified publications for their literature reviews on the basis of such rankings. This procedure seems to be more convenient and reliable.In this framework, a review of most commonly used scientometric principles and methods are presented in the... 

One of the fields, in which graph theory and computer science have been appleid,is architectural design. Some of these applications are presented in this thesis. This thesis has two parts, the first part peresents a systematic pathway for design of a floor plan when given the list of cells, the required dimensions of each cell and the matrix of required adjacencies between the cells. (Based on the article by Hashimshony(1988)) The second part is about the application of Evolutionary Algorithm (EA) in ar-chitucture. The architectural layout design problems, which is concerned with the finding of the best adjacencies between functional spaces among many possible ones under given constraints,... 

There are many fun plays and Sudoku is one of the most popular one. Although this game was published in 1979 for the first time as a puzzle in American magazine, it was published continuously from 1986 in Japan. This game has been known as an internationally popular game since 2005 and its national completion was held in Philadelphia. Sudoku had been considered as a hobby for several years before its mathematical importance was known in articles such as “Taking Sudoku seriously” and “The science behind Sudoku”. Using a lot of computer times, Gary McGuire et al in their article “There is no 16-Clue Sudoku: Solving the Sudoku Minimum Number of Clues Problem” found that the minimum number of... 

Since today graphs have many applications in different sciences including computer networks, in this thesis a number of graphs which have been widely used in computer networks, are studied.The graphs presented in this thesis are hypercubes, generalized hypercubes (Bhuyan and grawal, 1984), balanced hypercubes (Huang and Jie, 1992) and the star graphs (Sheldon and Akers, 1987), which are the generalization of the hypercubes and have similar properties.In general the graph which is considered for computer networks as an interconnection structure,should have small degrees of vertices, a small diameter, and a large number of paths between any two vertices. Thus in our studies these three... 

The Sudoku puzzle is played on a 9 × 9 grid which is divided into nine boxes each with 3 × 3 cells. Initially, some digits between 1 and 9 are given on the grid as the clues. Generalized Sudoku (a table of n2 by n2 that includes n2 blocks with n cells and in each of its rows, columns and blocks has all numbers from 1 to n2 appeared once and only once) can be considered too. Millions of people around the world are tackling one of the hardest problems in computer science without even knowing it. The first such puzzle appeared in the May 1979 edition of Dell P encil P uzzles and W ord Games and, according to research done by W ill Shortz, the crossword editor of the N ew Y ork T imes, was... 

A Latin square of order n, is an n n array filled with n different symbols, each occurring exactly once in each row and exactly once in each column.Two cells of a Latin square are independent when they are not in the same row, or in the same column and they are not the same symbol. If they are not independent, they are called dependent. A k-coloring of a Latin square is assigning k colors to its cells where no two dependent cells have the same color. The smallest k for which we have a k-coloring for a Latin square L with k colors is called chromatic number of L and we denote it by L). If we consider the Cayley table of an arbitrary finite group of order n, then we have a Latin square of... 

There is much interest in the theoretical study of the structure of hypercubes because their structure has played an important role in the development of parallel processing and is still quite popular and influential.This thesis will begin with an introduction to hypercubes and some of their problems. Then we study the following concepts: Arboricity (The minimum number of edge-disjoint forests into which a graph can be decomposed), Galactic number (similar to arboricity but such that each component of each forest is a star), Silver coloring (Given a set I of maximal independent vertices, a silver coloring with respect to I is such that every v 2 I sees every other colors in its closed... 

Finding necessary and sufficient conditions for a sequence to be graphic or in more general setting existence of factor with predscribed degree sequence of a general graph and also existence of a polynomial algorithm for finding this factor are classical problems of graph theory. These problems were discussed and solved between 1952 to 1972 by prominent mathematicians such as Erdős, Tutte, Edmonds et al. In this thesis we initially discuss most generalized forms of this problems namely Tutte’s f-factor and Lovasz’ (g; f)-factor theorems. Then some simple generalization of the special cases like Erdős-Gallai and Gale-Ryser theorems are investigated. And by using a form of Lovasz’ (g;... 

In any finite combinatorial configuration, it is possible to identify a subset which uniquely determines the structure of the configuration. Because of their numerous applications in combinatorics, they have a special importance and have been studied for instance in Latin squares, combinatorial designs, and graphs. These sets have been called critical sets in Latin squares, defining sets in combinatorial designs, and forcing sets in different concepts in graphs. In this thesis, we review some papers about critical sets in Latin squares. In Chapter 1, we review the necessary definitions and several theorems and lemmas, for studying critical sets. In Chapter 2, we review the use of Latin... 

Since critical and defining sets have been examined in various fields of combinatorics, including Latin squares and different branches of graph theory, we aim to present a portion of the obtained results con-cerning defining sets in diverse fields of graph theory, and provide a survey on them.In the present dissertation, we examine defining sets of vertex coloring,edge coloring, total coloring and forcing sets of orientation of graphs,perfect matching, hull and geodetic sets. Since complete graphs’ col-oring defining sets and Latin squares’ critical sets are closely related,we shall discuss them in a coloring framework. Furthermore, for dif-ferent graph families, including regular graphs,... 

Sudoku is an intellectual game based on mathematics. Sudoku achieved global fame in a short time because this game doesn’t need any mathematical operation like addition and multiplication and in theory, instead of numbers, lechers, shapes,colors, etc. could be placed.Sudoku which is a single-player game in generally the player should complete a partial n2 × n2 matrix M all of whose enteries are drawn from {1; 2,..., n2} by applying the rules: no number may appear more than once in any row, any column,or any of the nine ‘‘blocks’’. Actually blocks are n × n submatrixs with indices {na + 1, na + 2,…, na + n} × {nb + 1, nb + 2,….., nb + n} Actually, A Sudoku is A Latin square with one... 

In the year 1997, at 16th British Combinatorial Conference Cameron introduced a new concept called 2-simultaneous coloring. He used this concept to reformulate a conjecture of Keedwell on the existence of critical partial Latin squares of a given type. After that, other people studied special cases of this conjecture. They also studied the relation between this conjecture and other well-known conjectures in Combinatorics, such as Cycle Double Cover Conjecture (CDC). Finally, in 2004 the general case of this conjecture was proven. In first two sections, in addition to defining some fundamental concepts, we represent the relation between Latin trade and 2-simultaneous coloring. Afterwards, by... 

The concept of defining set in matching theory, recently has been taken into consideration by chemists and mathematicians because of several important applied problems in chemistry and mathematics. This concept is studied extensively in vari-ous areas of combinatorics and graph theory. The idea is to obtain total combinatorial structure of one object uniquely, based on some special information about it. Defin-ing set has been studied in various areas of graph theory like colouring, matching , orientations of graphs and etc. and many researches have been done in these areas. What we focus on, in this thesis is the defining set in matching theory that has been called “forcing set”. A minimum... 

Critical sets and defining sets in combinatorics have been attended by mathematics fans. These subjects have been debated since 1997 and a lot of researches have been done about them and a lot of articles have been published. But number of unsolved questions might be more than answered questions. In these years critical sets for Latin square and defining sets for vertex coloring have been attended and also enough researches about issues related to defining sets for edge coloring and total coloring have not been done. For these reasons we focus on these issues in this thesis. Issues like defining sets for edge coloring and total coloring in complete graphs, generalized Petersen graphs and also... 

Decomposition of a graph into it’s subgraphs is an important problem in Graph Theory and Combinatorics. In this thesis we investigate some papers and their results about the problem of the decomposition of a complete graph into 4−cycles. In chapter 1 we express some parts of a paper written by Bryant, Darryn, Horsley, Daniel, and Pettersson. After giving some definitions and notations about cycle systems and their spectrums, we use methods of the paper to give a proof for a theorem on the existence of a decomposition of complete graphs into 4−cycles. In the next three chapters we explain results of a paper written by Bryant, Darryn, Grannell, Mike, Griggs, Terry, and Mačaj, Martin. chapter 2...