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,B) where X is the vertex set of Kv and B is a collection of...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,B) where X is the vertex set of Kv and B 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 of B. 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.展开更多
This article provides an overview of some recent results and ideas relatedto the study of finite groups depending on the restrictions on some systems of theirsections.In particular,we discuss some properties of the la...This article provides an overview of some recent results and ideas relatedto the study of finite groups depending on the restrictions on some systems of theirsections.In particular,we discuss some properties of the lattice of all subgroups ofa finite group related with conditions of permutability and generalized subnormality for subgroups.The paper contains more than 30 open problems which were posed,atdifferent times,by some mathematicians working in the discussed direction.展开更多
Let λKv be the complete multigraph with v vertices and G 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,B) where X is ...Let λKv be the complete multigraph with v vertices and G 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,B) where X is the vertex set of Kv and B 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 of B. A packing (covering) design is said to be maximum (minimum) if no other such packing (cov...展开更多
A maximum （v, G, λ）-PD and a minimum （v, G, λ）-CD axe studied for 2 graphs of 6 vertices and 7 edges. By means of ＂difference method＂ and ＂holey graph design＂, we obtain the result： there exists a （v, Gi,...A maximum （v, G, λ）-PD and a minimum （v, G, λ）-CD axe studied for 2 graphs of 6 vertices and 7 edges. By means of ＂difference method＂ and ＂holey graph design＂, we obtain the result： there exists a （v, Gi, λ）-OPD （OCD） for v ≡ 2, 3, 4, 5, 6 （mod 7）, λ ≥ 1, i = 1, 2.展开更多
文摘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,B) where X is the vertex set of Kv and B 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 of B. 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.
文摘This article provides an overview of some recent results and ideas relatedto the study of finite groups depending on the restrictions on some systems of theirsections.In particular,we discuss some properties of the lattice of all subgroups ofa finite group related with conditions of permutability and generalized subnormality for subgroups.The paper contains more than 30 open problems which were posed,atdifferent times,by some mathematicians working in the discussed direction.
基金the National Natural Science Foundation of China (No.10671055)
文摘Let λKv be the complete multigraph with v vertices and G 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,B) where X is the vertex set of Kv and B 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 of B. A packing (covering) design is said to be maximum (minimum) if no other such packing (cov...
基金Supported by the National Natural Science Foundation of China (Grant No.10671055)
文摘A maximum （v, G, λ）-PD and a minimum （v, G, λ）-CD axe studied for 2 graphs of 6 vertices and 7 edges. By means of ＂difference method＂ and ＂holey graph design＂, we obtain the result： there exists a （v, Gi, λ）-OPD （OCD） for v ≡ 2, 3, 4, 5, 6 （mod 7）, λ ≥ 1, i = 1, 2.
