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arXiv 0906.3053math.CO
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广义连通性$κ_3(G)$的精确界

Sharp bounds for the generalized connectivity $κ_3(G)$

Shasha Li, Xueliang Li, Wenli Zhou

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18 pages
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摘要：

设$G$是一个非平凡的$n$阶连通图，$k$是一个具有$2\leq k\leq n$的整数.对于$G$的$k$顶点的集合$S$，令$κ(S)$表示$G$中的边不相交树$T_1,T_2,...,T_\ell$的最大数目$\ell$，使得对于具有$1\leq i,j\leq \ell$的每个不同整数对$i,j$，$V(T_i)\cap V(T_j)=S$。具有此属性的$G$中树的集合$\{T_1,T_2,...,T_\ell\}$称为连接$S$的树的内部不相交集。Chartrand等人将连通性的概念概括如下：$G$的$k$-$connectivity$，用$κ_k(G)$表示，由$κ_k(G)=$ min $\{κ(S)\}$定义，其中最小值取$V(G)$的所有$k$-子集$S$。因此是$κ_2(G)=κ(G)$，其中$κ(G)$是$G$的连通性。一般来说，$κ_k(G)$的研究非常困难。因此，本文主要对$κ_3(G)$进行研究。研究了图的连通度与3-连通度之间的关系。首先给出了一般图$G$的$κ_3(G)$的精确上、下界，并构造了两类分别达到上、下界的图。然后证明了如果$G$是连通平面图，则$κ(G)-1 \leq κ_3(G)\leq κ(G)$，并给出了达到这一界的几类图。最后证明了平面图$G$的$κ(G)=κ_3(G)$是否能在多项式时间内求解的问题。

英文摘要：

Let $G$ be a nontrivial connected graph of order $n$ and let $k$ be an integer with $2\leq k\leq n$. For a set $S$ of $k$ vertices of $G$, let $κ(S)$ denote the maximum number $\ell$ of edge-disjoint trees $T_1,T_2,...,T_\ell$ in $G$ such that $V(T_i)\cap V(T_j)=S$ for every pair $i,j$ of distinct integers with $1\leq i,j\leq \ell$. A collection $\{T_1,T_2,...,T_\ell\}$ of trees in $G$ with this property is called an internally disjoint set of trees connecting $S$. Chartrand et al. generalized the concept of connectivity as follows: The $k$-$connectivity$, denoted by $κ_k(G)$, of $G$ is defined by $κ_k(G)=$min$\{κ(S)\}$, where the minimum is taken over all $k$-subsets $S$ of $V(G)$. Thus $κ_2(G)=κ(G)$, where $κ(G)$ is the connectivity of $G$. In general, the investigation of $κ_k(G)$ is very difficult. We therefore focus on the investigation on $κ_3(G)$ in this paper. We study the relation between the connectivity and the 3-connectivity of a graph. First we give sharp upper and lower bounds of $κ_3(G)$ for general graphs $G$, and construct two kinds of graphs which attain the upper and lower bound, respectively. We then show that if $G$ is a connected planar graph, then $κ(G)-1 \leq κ_3(G)\leq κ(G)$, and give some classes of graphs which attain the bounds. In the end we show that the problem whether $κ(G)=κ_3(G)$ for a planar graph $G$ can be solved in polynomial time.

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