Some Results On Decomposition And Equipacking And Equipcovering Of Graphs

Packing and covering problems are very important and fundamental in graph the-ory which have great significance in physics,computer network and combinatorics optimization research.Packing and covering is a pair of dual conception.As an ac-tive branch of packing and covering,the equipacking and equicovering problems in graph theory stem from Caro and Schonheim characterizing P₃-decomposable graph and M₂-decomposable graph.A graph G is called H-equipackable if its every maxi-mal H-packing is also its maximum H-packing.A graph G is called H-equicoverable if its every minimal H-covering is also its minimum H-covering.In this paper,we study decomposing,equipacking and equcovering problems in graph theory.In chapter 1,we give a general survey and development on our research problems and list main results.In chapter 2,we mainly give the basic knowledge,basic definition,theorem and related symbols needed to study the problems in the paper.In chapter 3,we mainly use the transformation of the graph,the mathematical induction method to study the P₅-decomposable complete graph,complete bipartite graph,wheel graph and complete multipartite graph.In chapter 4,we study the P₄-equipackable complete graph,complete bipartite graph,fan graph and wheel graph by edge matrix.In chapter 5,we mainly research equicoverable problem,characterizing P₅-equico-verable graphs which contain cycles with length at least 5 and part of P_k-equicoverable graphs.