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- Type of Document: M.Sc. Thesis
- Language: Farsi
- Document No: 46566 (02)
- University: Sharif University of Technology
- Department: Mathematical Sciences
- Advisor(s): Akbari, Saieed
- Abstract:
- Graph coloring is a fundamental problem in Computer Science. Despite its status as a computationally hard problem, it is still an active area of research. Depending on the specific area of requirement or an pplication, there are several variants of coloring.One such variant of graph coloring is Acyclic Graph Coloring. As with the case of proper coloring of graphs, Acyclic coloring can either be on vertex set or the edge set of a graph. When such a coloring is performed on the edges of a graph, it requires that after a graph has been colored there should not exist any cycle that runs on edges that are colored by only two colors. Such a cycle is called a bichromatic cycle. When a coloring does not present itself with a bichromatic cycle, it is said to have an acyclic edge coloring. When such a coloring is performed on the vertices of a graph, a bichromatic cycle is said to be one that runs on the vertices that are colored by only two colors. In absence of such a bichromatic cycle, an edge coloring is said to be an acyclic vertex coloring of that graph. The smallest number of colors required to acyclically edge color a graph is called its acyclic chromatic number and is denoted by a′(G). Similarly, the smallest number of colors required to acyclically vertex colors a graph is called its acyclic vertex chromatic number and is denoted by a(G). There are several applications of acyclic coloring of a graph which include applications in coding theory as well as other theoretical problems. In addition to this, acyclic chromatic numbers are closely related to several other types to colorings of a graph each of which are applicable in several other fields of interest. A notable example among these is the star chromatic number of a graph. Star graphs are used to model star network topologies that have applications in computer networks and distributed computing. Also the problem of finding a′(G) is intimately related to other well known conjectures in graph theory which maybe the most noticeable one is Perfect 1-factorization conjecture which is an open problem yet.The problems of finding the acyclic chromatic number or acyclic edge chromatic number of a graph are known be NP-hard. Also, there is a very little knowledge about such a characteristic of a graph even for the most simple classes of graphs like complete graphs.
In this thesise first we will start with concept Perfect 1-factorization and after some preliminary definitions we will discuss about the relation between both concepts. Generally we will discuss about acyclic edge coloring of three family of graphs.
Regular graphs. Alon suggested a possibility that complete graphs of even order are the only regular graphs which require Δ + 2 colors to be acyclically edge colored. We will see that the answer is no and we will introduce some family of regular graphs which they need Δ+2 colors to be acyclicaly colored.Bipartite graphs. Bipartite graphs have always been of great value in graph theory and its applications, but there is few results about acyclic edge coloring of this graphs. In a section of this thesis we will show these few and interestig results. And it is noticeable to say the Lemma 11.3 and the Theorem 12.3 refer to the author. Cubic graphs. Many people has worked on acyclic edge colorin of cubic graphs since this consept arised and they have achieved some results.But finaly in 2010 every thing was done for acyclic edge coloring of cubic graphs. We wil discuss about them in more detail - Keywords:
- COLOURING ; Acyclic Matching ; Bipartite Graph ; Graph Coloring ; Perfect 1-Factorization
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