关于两个六点八边图的图设计

G-design with Six Vertices and Eight Edges

设Kv是一个v个点的完全图,G为Kv的一个不含孤立点的简单子图.Kv的一个G-设计,常记为(v,G,1)-GD,是指一个二元组(X,B),其中X为Kv的顶点集,B是Kv的一些子图(亦称为区组)构成的集合,使得每一个区组与G同构,且Kv的任何一条边恰在B的一个区组中出现.文章讨论了两个六点八边图G1和G2的图设计存在性问题,并证明了(v,Gi,1)-GD(i=1,2)存在的必要条件v≡0,1(mod 16)且vE 16也是充分的.

Let Kv be a complete graph with v vertices, and G be a simple subgraph without isolate vertices of Kv. A G - design of Kv, denoted by （ v, G, 1 ） - GD, is a pair （ X, B）, where X is the vertex set of K, and B is the collection of subgraphs （called blocks） of Kv, such that each block is isomorphic to G, and any edge in Kv occurs in exacdy one block. In this paper, we discussed the existence problem of two graphs G1G2, each of which is with six vertices and eight edges, and showed that （v, Gi, 1） - GD（ i = 1, 2） exist if and only if v=0, 1 （rood 16） and v≥16.