This book is based on lecture notes written for summer graduate schools at Nankai University and Shandong University in 2005.The selection of the material was of course heavily influenced by our personal interests，as well as the limitations of space，while trying to cope with the recent development in the areas not covered by any known book.The book is primarily aimed at researchers and graduate students in graph theory.However，most of the material discussed is accessible to anyone with an undergraduate level understanding of mathematics.Our main source of materials is from three sources:research articles，textbooks and survey papers.

Graph Factors and Matching Extensions deals with two important branches of graph theory  factor theory and extendable graphs.Due to the mature techniques and wide ranges of applications，factors and matchings become useful tools in investigation of many theoretical problems and practical issues.This book includes basic techniques，theoretical foundations，as well as recent advances in the field.It also discusses open problems and conjectures to inspire the readers.

This book is primarily aimed at researchers and graduate students in graph theory and combinatorics in general.However，all the necessary prerequisites are developed from scratch and should be accessible to upper level undergraduate students with prior knowledge of graph theory.

1 Matchings and Perfect Matchings

 1.1 Definitions and terminologies

 1.2 Matchings in bipartite graphs and augmenting path

 1.3 Matchings in non-bipartite graphs

 1.4 Sufficient conditions for 1-factors

 1.5 Gallai-Edmonds Structure Theorem

 1.6 Number of 1-factors

2 Degree Constraint Factors

 2.1 Characterizations of factors

 2.2 Factors in bipartite graphs

 2.3 Factors with special properties

 2.4 L-factor

3 Factors and Graphic Parameters

 3.1 Toughness and k-factors

 3.2 Toughness and [a，b]-factors

 3.3 Binding number and factors

 3.4 Connectivity and factors

 3.5 Other parameters and existence of factors

4 Component Factors and Connected Factors

 4.1 Star factor

 4.2 Path and cycle factors

 4.3 El-Zahar's Conjecture and other component factors

 4.4 Connected [a，b]-factors

 4.5 Connected (g，f)-factors

 4.6 Generalized trees

5 Elementary Graphs and Decomposition Theory

 5.1 Elementary graphs and 1-extendable graphs

 5.2 Ear decomposition

 5.3 Minimal graphs and more decompositions

 5.4 Bricks and optimal ear decomposition

6 k-Extendable Graphs and n-Factor-Critical Graphs

 6.1 Characterizations and basic properties

 6.2 Equivalence and recursive relationships

 6.3 Matching extension and graphic parameters

6.3.1 Matching extension and forbidden subgraphs

6.3.2 Matching extension and toughness

6.3.3 Matching extension in planar graphs and surfaces

6.3.4 Matching extension，degree sum and closure operations

6.3.5 Matching extension and product of graphs

6.3.6 Matching extension and other parameters

 6.4 Extendability of symmetric graphs

7 Extremal k-Extendable Graphs and Generalizations

 7.1 Maximal and minimal k-extendable graphs

 7.2 Generalization of matching extension

 7.3 Variations of graph extension

8 Fractional Factors of Graphs

 8.1 Fractional matchings

 8.2 Fractional (g，f)-factors

 8.3 Parameters and fractional factors of graphs

 8.4 Maximum and minimum fractional (g，f)-factors

 8.5 Connected fractional factors

Index

References