Research On The Existence Of Vertex-disjoint Copies Of Small Order Subgraphs In Claw-free Graphs

The emergence and development of graph theory has experienced two hundred years of history.It is an important branch of combinatorial mathematics.In this paper,we considered simple,finite,undirected graphs with no multiple edges and no loops.Claw-free graph is one of those graphs.A graph is called claw-free graph if it does not contain induced subgraph isomorphic to a claw.let K₄^- denote the graph which obtained from K4 by removing exactly one edge and let K1,t denote the star of order t+1.At the same time,let k be an integer with k?2.In this paper,the existence of vertex-disjoint copies of small order subgraphs in claw-free graphs are discussed.The main contents are as follows.First,we introduce some notations and terminology,the history and the progress of the problem we study.Second,we prove that every claw-free graph with order n?13k-12 and ?5?G??4 contains at least k vertex-disjoint copies of K1,4.Third,we prove that every claw-free graph with order n?12k-11 and ??G??5 contains at least k vertex-disjoint copies of K₄^-.The requirement of number five is necessary.Finally,in the each chapter,we list problems for future research and discussions.