λKv的最大K2,3填充设计和最小K2,3覆盖设计

对于一个有限简单图G,λKv的G-设计(G-填充,G-覆盖),记为(v,G,λ)-GD((v,G,λ)-PD,(v,G,λ)-CD),是一个(X,B),其中X是Kb的顶点集,B是Kv的子图族,每个子图(称为区组)均同构于G,且Kv中任一边都恰好(最多,至少)出现在B的λ个区组中.一个填充(覆盖)设计称为是最大(最小)的,如果没有其它的这种填充(覆盖)设计具有更多(更少)的区组.本文对于λ>1确定了(v,K2,3,λ)-GD的存在谱,并对任意λ构造了λKv的最大K2,3-填充设计和最小K2,3-覆盖设计.

Maximum K2,3-Packing Designs and Minimum K2,3-Covering Designs of λKv

Let G be a finite simple graph. A G-design (G-packing design, G-covering design)of λKv, denoted by (v, G, λ)-GD ((v, G, λ)-PD, (v, G, λ)-CD), is a pair (X, β) where X is the vertex set of Kv andβ is a collection of subgraphs of Kv, called blocks, such that each block is isomorphic to G and any two distinct vertices in Kv are joined in exactly (at most, at least)λ blocks of3. A packing (covering) design is said to be maximum (minimum) if no other such packing (covering) design has more (fewer) blocks. In this paper, we determine the existence spectrum for the K2,3-designs of λKv, λ＞ 1, and construct the maximum packing designs and the minimum covering designs of λKv with K2,3 for any integer λ.