几类7点7边图的图设计与最优填充

Decompositions and Optimal Packings of K_v into the Graphs with Seven Vertices and Seven Edges

设λKv 是λ重v点完全图 ,G是无孤立点的有限简单图 .将G设计 (G填充 )记作 (v ,G ,λ)GD( (v ,G ,λ)PD)是指一个序偶 (X ,B) ,其中X是完全图Kv 的顶点集 ,B是Kv 中同构于G的子图 (区组 )的集合 ,使得Kv 中每条边恰好 (至多 )出现在B的λ个区组中 .讨论了 3类 7点 7边图Gi(i=1,2 ,3)的图设计及最优填充问题 ,并给出了 (v ,Gi,1)GD及 (v ,Gi,1)OPD (i=1,2 ,3)存在的谱 .

Let λK v be the complete multigraph with v vertices,G be a finite simple graph.A Gdesign(Gpacking design) of λK v,denoted by (v,G,λ)GD ((v,G,λ)PD) is a pair (X,B),where X is the vertex set of K v and B is a collection of subgraphs of K v,called blocks,such that each block is isomorphic to G and any two distinct vertices in K v are joined in exactly (at most) λ blocks of B.The G i-designs and G i-optimal packings is discussed,where G i(i=1,2,3) has seven vertices and seven edges,and solved the existence spectrum of (v,G i,1)GD and (v,G i,1)OPD,i=1,2,3.