On The Extendability Of Cayley Graphs Of A Kind Of Clifford Semi-groups

Let G be a semi-graph and C (?) G.We define the Cayley graph Cay(G,C) as follows:G is the vertex set and x1x2, x1,x2∈G, is an edge in Cay(G,C) if there exists an element c∈C such that x2=cx1=cx2.Let S=(G∪F,(?)) be a Clifford semi-groups with G, F are Abelian groups,(?) is a group monomorphism from G to F. Let C1and C2are subsets of G and F with C1=C1-1,C2=C2-1,respectively.We denote that C=C2∪C1. Define Cayley graph Г=Cay(S,C) to have vertex set S, and x1x2G∈,(Г) if and only if one of the following three possibilities occurs:(1) x1,x2∈G, and c∈C1such that x2=cx1or x1=cx2;(2) x1,x2∈F, and c b C2∪(?)(C1) such that x2=cx1ir x1=cx2;(3) x1∈G,x2∈F, and c∈C2such that x2=c(?)(x1).A connected even graph Г with at least2n+2vertices is said to be n-extendable if every matching of size n in Г can be extended to a perfect matching. A connected odd graph Г is said to be n1/2-extendable if for every vertex υ of Г the graph Г-υ is n-extendable. The aim of this paper is to study the extendability of Cayley graphs of a kind of Clifford semi-groups S=(G∪F,(?)). This dissertation consists of two chapters.In the first chapter,some research backgrounds and preliminaries are intro-duced.In chapter2,we studied the extendability of Cay(S,C).We proved that Cay(S,C)are1-extendable when|G|+|F|is even;and characterized the11/2-extendability of Cay(S,C) when|G|+|F|is odd,2-extendability of Cay(S,C) when|G|+|F|is even.