广义q-树的群连通度(英文)  

Group Connectivity of Generalized q-trees

作　　者：郝荣霞[1] 李德明[2] 李赵祥[3] 

机构地区：[1]北京交通大学数学系,北京100044 [2]首都师范大学数学系,北京100037 [3]中央民族大学数学系,北京100081

出　　处：《数学进展》2012年第6期693-697,共5页Advances in Mathematics（China）

基　　金：Supported by NSFC(No.10871021 and No.11171020)

摘　　要：设G是无向图,A是加法交换群,且A*=A-0.如果G有一个定向D(G),对任何满足∑_(v∈V(G))b(v)=0的函数b:V(G)→A,都存在函数f:E(G)→A*使得在每个顶点v∈V(G),从v发出的所有边上的f总值减去进入v的所有边上的f总值恰等于b(v),则称G是A-连通的.群连通数为:A_g(G)=min{n:对任何满足|A|≥n的群A,G是A-连通的}.令q是正整数,广义q-树的定义是按下列递推形式给出的:最小的广义q-树是阶为q的完全图K_q;阶为n+1的广义q-树是由阶为n的广义q-树通过增加一个新顶点和连接此点与阶为n的广义q-树中任意给定的q个顶点得到的.本文对广义q-树G(q≥2)考察Λ_g(G),证明了如果G是阶n≥3的广义2-树或阶为n∈{3,4}的广义3-树或阶为4的广义4-树,则Λ_g(G)=4;如果G是阶为n≥5的广义q-树(q≥3),则A_g(G)=3.Let G be an undirected graph, A be an （additive） abelian group and A^＊ = A - 0. A graph G is A-connected if G has an orientation D（G） such that for every function b,： V（G）→ A satisfying ∑v∈v（a） b（v） = 0, there is a function f ： E（G） → A^＊ such that at each vertex v∈ V（G）, the amount of f values on the edges directed out from v minus the amount of f values on the edges directed into v equals b（v）. The group connectivity number Ag（G） = min{n ： G is A-connected for every abelian group A with ｜A｜≥n}. Let q be a positive integer. The generalized q-trees are defined by recursion： the smallest generalized q-tree is the complete graph Kq -with order q, and a generalized q-tree with order n ＋ 1 where n ≥ q is obtained by adding a new vertex adjacent to q arbitrarily selected vertices of a generalized q-tree with order n. In this paper, we investigate Ag（G） for generalized q-trees G with q ≥ 2. We show that if G is a generalized 2-tree with order n 〉 3 or generalized 3-tree with order n E {3, 4} or generalized 4-tree with order 4, then A9 （G） = 4, and if G is a generalized q-tree for q ≥ 3 with order n ≥ 5, then A, （G） = 3.

关 键 词：交换群 A-连通 群连通性 

分 类 号：O157.5[理学—数学]