This work is our selected results of research on graph partitioning and matching problems in the field of theoretical computer science and structural graph theory in recent years. After an introductory chapter， the reader will find six chapters， each of which is written as a self-contained content. In the first part of the work， Chapter 2 through 4， we concentrate on the complexity， inapproximability， approximation algorithms and on-line algorithms of some graph vertex partitioning problems. In the second part of the work， Chapter 5 through 7， we focus on the structural properties of some graph problems related to matching wluch can be regarded as edge partitioning problems. We refer to the listed chapters for the details of the results.
Chapter 1 contains a short general introduction to the topics of the book and gives an overview of the main results， together with some motivation and connections to and relationships with older results. Specific terminology and notation can be found just before each of the topic8.
In Chapter 2， we first investigate the computational complexity of problems of determining the minimum number of monochromatic cliques or rainbow cycles that， respectively， partition the vertex set V(G) of a graph G. We show that the minimum monochromatic clique partition problem is APX-hard on K4 -free graphs and monochromatic-K4 -free graphs， and APX-complete on monochromatic-K4 - free graphs in which the size of a max：imum monochromatic clique is bounded by a constant. We also show that the minimum rainbow cycle partition problem is NP-complete， even if the input graph G is triangle-free. Moreover， for the weighted version of the minimum monochromatic clique partition problem on monochromatic-K4 -free graphs， we derive an approximation algorithm with (tight) approximation guarantee In |V (G)|+1.
This work is our selected results of research on graph partitioning and matching problems in the field of theoretical computer science and structural graph theory in recent years. After an introductory chapter， the reader will find six chapters， each of which is written as a self-contained content. In the first part of the work， Chapter 2 through 4， we concentrate on the complexity， inapproximability， approximation algorithms and on-line algorithms of some graph vertex partitioning problems. In the second part of the work， Chapter 5 through 7， we focus on the structural properties of some graph problems related to matching wluch can be regarded as edge partitioning problems. We refer to the listed chapters for the details of the results.
Chapter 1 contains a short general introduction to the topics of the book and gives an overview of the main results， together with some motivation and connections to and relationships with older results. Specific terminology and notation can be found just before each of the topic8.
In Chapter 2， we first investigate the computational complexity of problems of determining the minimum number of monochromatic cliques or rainbow cycles that， respectively， partition the vertex set V(G) of a graph G. We show that the minimum monochromatic clique partition problem is APX-hard on K4 -free graphs and monochromatic-K4 -free graphs， and APX-complete on monochromatic-K4 - free graphs in which the size of a max：imum monochromatic clique is bounded by a constant. We also show that the minimum rainbow cycle partition problem is NP-complete， even if the input graph G is triangle-free. Moreover， for the weighted version of the minimum monochromatic clique partition problem on monochromatic-K4 -free graphs， we derive an approximation algorithm with (tight) approximation guarantee In |V (G)|+1.
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