K_(2,3)的多重多部图设计的构造

λKn(t)是一个λ重完全多部图,G为一个不带孤立点的简单图.所谓的图设计G-HDλ(tn)是一个序偶(X,B),其中X是Kn(t)的顶点集,B为λKn(t)的一些子图(亦称为区组)构成的集合,使得任一区组均与图G同构,且λKn(t)的任意2个不同点组成的边恰在B的λ个区组中出现.本文讨论了G=K2,3的完全多部图设计存在性问题,证明了存在G-HDλ(tn)当且仅当λn(n-1)t2≡0(mod12),n≥2,nt≥5且(n,,λt)≠(9,1,1),(12,1,1),(3,1,2),(4,1,2).

Construction of λ-Fold Complete Multipartite Graph Design of K2,3

Let λKn(t) be a complete multipartite multigraph,G be a finite simple graph.So-called graph design G-HDλ(tn),is a pair of(X,B),where X is the vertex set of Kn(t) and B is collection of sub-graphs(called blocks) of λKn(t),such that each block is isomorphic to G and any edge in λKn(t) occur in exactly λ block of B.The existence of λ-fold complete multipartite graph design is discussed where G=K2,3.The following existence is obtained: G-HDλ(tn) exists if and only if λn(n-1)t2≡0(mod12),n≥2,nt≥5 and(n,λ,t)≠(9,1,1),(12,1,1),(3,1,2),(4,1,2).