On The Existence Of Vertex-Disjoint Subgraphs Of Order Four In 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 are 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. let G be a claw-free graph with order n and minimum degree δ, we present the number of vertex-disjoint K4- in G by virtue of δ and n.A multigraph is a nonempty set of vertices, every tow of which are joined by a finite number of edges. A cycle of length four is called a quadrilateral and a multigraph is called standard if every edge in it has multiplicity at most two with no loops,Record as a M. let D be a directed graph of order 4k and where k is a positive integer. Suppose that the minimum degree of D is at least 6k-2, then D contains k vertex-disjoint directed quadrilaterals with only one exception.This paper is divided into four chapters.In chapter 1, we introduce some notations and terminology, the history and the progress of the prob-lem of we study.In chapter 2, we consider Let G be a claw-free graph with order n and minimum degree δ≥5,then G contains at least F’(n,δ)=(δ-4/7δ-8)n vertex-disjoint K4-.In chapter 3, we consider Let M is a standard multigraph of order 4k, where k is a positive inte-ger.Suppose that δ(M)≥6k-2, then M contains k vertex-disjoint Q74, unless M ∈{D*, F*}.Furthermore, in the each chapter, we list problems for future research and discussions.