arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 1808.10074math.CO
收藏

某些正则图的广义连通度

The generalized connectivity of some regular graphs

Shu-Li Zhao, Rong-Xia Hao

备注：
19 pages, 6 figures
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

图$G$的广义$k$ -连通度$κ_{k}(G)$是一个度量网络$G$连接$G$中任意$k$顶点的可靠性的参数，对于一般图$G$，它被证明是NP-完全的.令$S\subseteq V(G)$和$κ_{G}(S)$表示$G$中边不相交树$T_{1}, T_{2}, \cdots, T_{r}$的最大数目$r$，使得对于任何$i, j \in \{1, 2, \cdots, r\}$和$i\neq j$，$V(T_{i})\bigcap V(T_{j})=S$。对于具有$2\leq k\leq n$的整数$k$，图$G$的{ \em广义$k$ -连通度}定义为$κ_{k}(G)= min\{κ_{G}(S)|S\subseteq V(G)$和$|S|=k\}$。本文研究了递归构造的一般$m$ -正则和$m$ -连通图$G_{n}$的广义$3$ -连通性，得到了$κ_{3}(G_{n})=m-1$，它达到了Li { \em et al.}[Discrete Mathematics 310（2010）2147-2163]给出的$κ_{3}(G)$的上界.关于$G=G_{n}$作为主要结果的应用，可以直接得到交替群图$AG_{n}$、$k$ -元$n$ -立方体$Q_{n}^{k}$、分裂星形网络$S_{n}^{2}$和泡排序星形图$BS_{n}$等著名网络的广义$3$ -连通性.

英文摘要：

The generalized $k$-connectivity $κ_{k}(G)$ of a graph $G$ is a parameter that can measure the reliability of a network $G$ to connect any $k$ vertices in $G$, which is proved to be NP-complete for a general graph $G$. Let $S\subseteq V(G)$ and $κ_{G}(S)$ denote the maximum number $r$ of edge-disjoint trees $T_{1}, T_{2}, \cdots, T_{r}$ in $G$ such that $V(T_{i})\bigcap V(T_{j})=S$ for any $i, j \in \{1, 2, \cdots, r\}$ and $i\neq j$. For an integer $k$ with $2\leq k\leq n$, the {\em generalized $k$-connectivity} of a graph $G$ is defined as $κ_{k}(G)= min\{κ_{G}(S)|S\subseteq V(G)$ and $|S|=k\}$. In this paper, we study the generalized $3$-connectivity of some general $m$-regular and $m$-connected graphs $G_{n}$ constructed recursively and obtain that $κ_{3}(G_{n})=m-1$, which attains the upper bound of $κ_{3}(G)$ [Discrete Mathematics 310 (2010) 2147-2163] given by Li {\em et al.} for $G=G_{n}$. As applications of the main result, the generalized $3$-connectivity of many famous networks such as the alternating group graph $AG_{n}$, the $k$-ary $n$-cube $Q_{n}^{k}$, the split-star network $S_{n}^{2}$ and the bubble-sort-star graph $BS_{n}$ etc. can be obtained directly.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑