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圖上獨立圈若干問題的結構參數 版權信息
- ISBN：9787030649928
- 條形碼：9787030649928 ; 978-7-03-064992-8
- 裝幀：一般膠版紙
- 冊數：暫無
- 重量：暫無
- 所屬分類：>
圖上獨立圈若干問題的結構參數 內容簡介
The study of graph theory started over two hundreds years ago. The earliest known paper is due to Euler（1736） about the seven bridges of Korugsberg. Since 1960s， graph theory has developed very fast and numerous results on graph theory sprung forth. There are many nice and celebrated problems in graph theory， such as Hamiltonian problem， four-color problem， Chinese postman problem， etc. Moreover， graph theory is widely applied in chemistry， computer science， biology and other disciplines. As a subfield in discrete mathematics， graph theory has attracted much attention from all perspectives. All graphs are considered only finite， simple， undirected graphs with no loops and no multiple edges. Let G be a graph. The Hamiltonian cycle problem is one of the most well-known problems in graph theory. A cycle which contains every vertex of G is called a Hamiltonian cycle. A cycle is called a chorded cycle if this cycle contains at least one chord. A k-factor in a graph G is a spanning k-regular subgraph of G， where k is a positive integer. There exists many interesting results about the existence of k-factor， by applying Tutte's Theorem， however， we mainly focus on the existence of 2-factor throughout this thesis. Clearly， a Hamiltonian cycle is a 2-factor with exactly one component. From this point of view， it is a more complex procedure to find the condition to ensure the existence of 2-factor in a given graph. The most usual technique to resolve 2-factor problems is to find a minimal packing and then extend it to a required 2-factor. The book is concerned with structural invariants for packing cycles in a graph and partitions of a graph into cycles， i.e.， finding a prescribed number of vertex-disjoint cycles and vertex-partitions into a prescribed number of cycles in graphs. It is well-known that the problem of determining whether a given graph has such partitions or not， is NP-complete. Therefore， many researchers have investigated degree conditions for packing and partitioning. This book mainly focuses on the following invariants for such problems： minimum degree， average degree （also extremal function）， degree sum of independent vertices and the order condition with minimum degree.
圖上獨立圈若干問題的結構參數 目錄
Contents
Preface
Notations
Chapter 1 Introduction and Main Results
1.1 Basic concepts and definitions
1.2 Invariants for 2-factors in graphs
1.3 Degree condition for 2-factors in bipartite graphs
1.4 Invariants for vertex-disjoint cycles in graphs
1.5 Invariants for vertex-disjoint cycles with constraints
1.5.1 Degree conditions for vertex-disjoint cycles containing prescribed elements
1.5.2 Degree conditions for vertex-disjoint cycles with length constraints in digraphs
1.5.3 Degree conditions for vertex-disjoint cycles with length constraints in tournaments
1.6 Outline the main results
Chapter 2 Nei***orhood Unions for Disjoint Chorded Cycles in Graphs
2.1 Introduction
2.2 Basic induction
2.3 Proof of Theorem 2.
Chapter 3 Vertex-Disjoint Double Chorded Cycles in Bipartite Graphs
3.1 Introduction
3.2 Lemmas
3.3 Proof of Theorem 3.
Chapter 4 2-Factors with Specified Elements in Graphs
4.1 2-Factors with chorded quadrilaterals
4.1.1 Lemmas
4.1.2 Proof of Theorem 4.
4.2 2-Factors Containing Specified Vertices in A Bipartite Graph
4.2.1 Lemmas
4.2.2 Proof of Theorem 4.
4.2.3 Proof of Theorem 4.
4.2.4 Discussion
Chapter 5 Packing Triangles and Quadrilaterals
5.1 Introduction and terminology
5.2 Lemmas
5.3 Proof of Theorem 5.
Chapter 6 Extremal Function for Disjoint Chorded Cycles
6.1 Extremal function for disjoint cycles in graphs
6.2 Proof of Theorem 6.
6.3 Basic Lemmas
6.4 Proof of Theorem 6.
6.5 Proof of Theorem 6.
6.6 Extremal function for disjoint cycles in bipartite graphs
6.7 Lemmas
6.8 Proof of Theorem 6.
6.9 Proof of Theorem 6.
6.10 Discussion
Chapter 7 Disjoint Cycles in Digraphs and Multigraphs
7.1 Disjoint cycles with di.erent lengths in digraphs
7.2 Disjoint quadrilaterals in digraphs
7.2.1 Introduction
7.2.2 Preliminary Lemmas
7.2.3 Proof of Theorem 7.
Chapter 8 Vertex-Disjoint Subgraphs with Small Order and Small Minimum Degree
8.1 Disjoint F in K1;4-free graphs with minimum degree at least four
8.1.1 Preparation for the proof of the Theorem 8.
8.1.2 Proof of the Theorem 8.
8.2 Disjoint K.4 in claw-free graphs with minimum degree at least five
8.2.1 Definition of several graphs
8.2.2 Preparation for the proof of the Theorem 8.
8.2.3 Proof of the Theorem 8.
8.2.4 Discussion
References
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