图上独立圈若干问题的结构参数(英文版)

图的点不交圈问题是有名的哈密尔顿圈及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条件的点不交圈结构参数，主要包括Dirac型最小度参数、极值参数以及邻域并参数，本书得到的这些参数大多是优选可能的。本书的主要结果如下：第一章引言部分，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，确定了图上有指定个数点不交弦圈的邻域并条件，这个界是优选可能的；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型最小度条件；第四章主要研究了图上有圈长以及指定顶点要求的两类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型条件下图的最小阶数；第六章主要确定了一般图中包含指定个数独立偶长圈以及二分图中包含指定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最小出度条件以及标准多重图中的Dirac型度条件；最后，第八章研究了最小度至少为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 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 Neighborhood 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