一类六点八边图的图设计

G-design for a Class of Graphs 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.文章讨论了一类六点八边图中尚未解决的3个图Gi(i=1,2,3)的图设计存在性问题,并证明了(v,Gi,1)-GD(i=1,2,3)存在的必要条件v≡0,1(mod16)且v16也是充分的.从而给出了这类六点八边图图设计存在的完全解.

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 K, occurs in exactly one block. In this paper, we discuss the existence problem of three graphs Gi（ i = 1, 2, 3）, which are unsolved in a class of graphs with six vertices and eight edges, and show that （v, Gi, 1） - GD（ i= 1, 2, 3） exists if and only if v≡0, 1 （rood 16） and v≥ 16 . Finally, the complete solutions of graph design for this class of graphs are given.