On The Extendability Of Semi-Cayley Graphs Of Abelian Groups

A graph SC(G)is said to be a semi-Cayley graph of a group G if it admits G as a semriregular automorphism group with two orbits(of equal size). Let A,B,C be subsets of a Abelian group G such that A=—A,B=—B and0∈A∪B. Dcfinc the grapgh SC(G:A,B,C)of the Abelian group G to have vertex set G×(0,1),and witll vertices(g,i),(h,j) adjacent if and only if one of the following three possibmties occurs:(1)i=j=0and h-g∈A；(2)i=j=0and h-g∈B；(3)i=j=0and h-g∈C；Then SC(G；A,B,C)is a sen-ii-Cayley graph of the group G.Let connected even graph (?) with at least2n+2vertices adnlitting a per—flect matching M. Then (?) is said n-extendable if every matching of size n in r can be extend to a perfect matching. The,I-graph I(n,j,k)is a graph with vertex set V(I)={ui,ui,|O≤i≤n-1)and edge set E(I){uiti+j,uiui,uiui+k|O≤i≤n-1).I-graph I(n,j,k)is a certain semi-cayley graph sc(zn；{±j},{±k},{0})0f integer addition groups.[13] and [7]character—ized the extendability of senli—Cayley graphs SC(G；(?),(?),C) and SC(G；A,A,C), respectivel tWe study the k-extendabiljty of Semi-Cayley graphs SC(G；A,B,C) of0ther cases(k≤2).ThUS we solve the k-extendability0f senli-Cayley graphs SC(G；A,JE},C)completely(k≤2).This dissertation consists of three ehapters.In the first chapter,some research backgrounds and preliminaries are intro-duced.In chapter2,we proved that all,I-graph I(n,j,k)are1-extendability and characterized the2-extendability of,I-grah. In chapter3, we proved that all semi-Cayley graphs SC(G; A,B,C) are1-extendability, and characterized the2-extendability of semi-Cayley graphs SC(G; A,B,C).