图上独立圈若干问题的结构参数（英文）

　　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 Tuttes 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.

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