[Original Authorization] Structural Parameters of Several Problems in the Independent Circle on the Figure (English) 9787030**9928

图的点不交圈问题是著名的哈密尔顿圈及2-因子问题的推广，具有重要的理论价值和实际应用价值，是图论研究的核心问题之一。本书主要研究了图上有限制条件的点不交圈结构参数，主要包括Dirac型*小度参数、极值参数以及邻域并参数，本书得到的这些参数大多是*好可能的。本书的主要结果如下：**章引言部分，主要介绍常用的图论术语和基本引理，以及本书的主要结果概述；第二章，确定了图上有**个数点不交弦圈的邻域并条件，这个界是*好可能的；第三章基于构造性证明，给出了均衡二部图中点不交双弦圈的Dirac型*小度条件；第四章主要研究了图上有圈长限制以及**顶点要求的两类2-因子问题，给出了Ore型界；第五章探讨了图上点不交三角形和四边形的填装问题，确定了在Ore型条件下图的*小阶数；第六章主要确定了一般图中包含**个数独立偶长圈以及二分图中包含**个数独立弦圈的边极值参数条件；第七章主要研究了有向图中包含独立圈的*小出度条件以及标准多重图中的Dirac型度条件；*后，第**研究了*小度至少为4和5的某些特殊图上含小阶子图的点数极值条件。本书详细介绍了上述问题的提出，发展过程以及完整的理论证明，并且提出了一些供进一步研究的问题。

• Contents
• Preface
• Notations
• Chapter 1 Introduction and Main Results 1
• 1.1 Basic concepts and definitions 1
• 1.2 Invariants for 2-factors in graphs 3
• 1.3 Degree condition for 2-factors in bipartite graphs 8
• 1.4 Invariants for vertex-disjoint cyc*** in graphs 10
• 1.5 Invariants for vertex-disjoint cyc*** with constraints 15
• 1.5.1 Degree conditions for vertex-disjoint cyc*** containing prescribed elements 15
• 1.5.2 Degree conditions for vertex-disjoint cyc*** with length constraints in digraphs 20
• 1.5.3 Degree conditions for vertex-disjoint cyc*** with length constraints in tournaments 21
• 1.6 Outline the main results 23
• Chapter 2 Neighborhood Unions for Disjoint Chorded Cyc*** in Graphs 26
• 2.1 Introduction 26
• 2.2 Basic induction 28
• 2.3 Proof of Theorem 2.4 28
• Chapter 3 Vertex-Disjoint Double Chorded Cyc*** in Bipartite Graphs 38
• 3.1 Introduction 38
• 3.2 Lemmas 41
• 3.3 Proof of Theorem 3.5 58
• Chapter 4 2-Factors with Specified Elements in Graphs 68
• 4.1 2-Factors with chorded quadrilaterals 68
• 4.1.1 Lemmas 69
• 4.1.2 Proof of Theorem 4.2 74
• 4.2 2-Factors Containing Specified Vertices in A Bipartite Graph 84
• 4.2.1 Lemmas 86
• 4.2.2 Proof of Theorem 4.6 91
• 4.2.3 Proof of Theorem 4.7 96
• 4.2.4 Discussion 99
• Chapter 5 Packing Triang*** and Quadrilaterals 100
• 5.1 Introduction and terminology 100
• 5.2 Lemmas 102
• 5.3 Proof of Theorem 5.3 108
• Chapter 6 Extremal Function for Disjoint Chorded Cyc*** 123
• 6.1 Extremal function for disjoint cyc*** in graphs 123
• 6.2 Proof of Theorem 6.3 127
• 6.3 Basic Lemmas 131
• *.* Proof of Theorem 6.5 135
• 6.5 Proof of Theorem 6.9 143
• 6.6 Extremal function for disjoint cyc*** in bipartite graphs 151
• 6.7 Lemmas 152
• 6.8 Proof of Theorem 6.12 156
• 6.9 Proof of Theorem 6.13 157
• 6.10 Discussion 1**
• Chapter 7 Disjoint Cyc*** in Digraphs and Multigraphs 166
• 7.1 Disjoint cyc*** with di.erent lengths in digraphs 166
• 7.2 Disjoint quadrilaterals in digraphs 177
• 7.2.1 Introduction 177
• 7.2.2 Preliminary Lemmas 179
• 7.2.3 Proof of Theorem 7.2 181
• Chapter 8 Vertex-Disjoint Subgraphs with Small Order and Small Minimum Degree 195
• 8.1 Disjoint F in K1;4-free graphs with minimum degree at least four 195
• 8.1.1 Preparation for the proof of the Theorem 8.4 198
• 8.1.2 Proof of the Theorem 8.4 211
• 8.2 Disjoint K.4 in claw-free graphs with minimum degree at least five 214
• 8.2.1 Definition of several graphs 215
• 8.2.2 Preparation for the proof of the Theorem 8.7 216
• 8.2.3 Proof of the Theorem 8.7 227
• 8.2.4 Discussion 227
• References 229