arXiv 1106.44112011-06-22math.CO

关于广义连通度kappa_3= 2的图的最小尺寸的注记

Note on the minimal size of a graph with generalized connectivity kappa_3= 2

Shasha Li, Xueliang Li, Yongtang Shi

备注：: 7 pages

摘要：

图$G$的广义$k$ -连通度$κ_{k}(G)$的概念是由Chartrand等人近年来提出的.在我们早期的论文中，开始了这个图参数的极值理论。我们确定了具有$κ_{3}= 2$的$n$阶图的最小边数，即，对于阶为$n$、大小为$e(G)$且具有$κ_{3}(G)= 2$的图$G$，通过构造一类图证明了$e(G)\geq (6/5)n$，且下界是尖锐的，仅对$n\equiv 0 \ (mod \ 5)$和$n\neq 10$ .在本文中，我们改进的下界$\lceil(6/5)n\rceil$。此外，我们表明，对于所有的$n\geq 4$，但$n= 9, 10$，总是存在一个图的顺序$n$与$κ_{3}= 2$的大小达到下界$\lceil(6/5)n\rceil$。而对于$n= 9, 10$，我们给出例子来表明$\lceil(6/5)n\rceil+1$是最好的可能下界。这给出了一个清晰的图片上的最小尺寸的顺序$n$与广义连通$κ_{3}= 2$。

英文摘要：

The concept of generalized $k$-connectivity $κ_{k}(G)$ of a graph $G$ was introduced by Chartrand et al. in recent years. In our early paper, extremal theory for this graph parameter was started. We determined the minimal number of edges of a graph of order $n$ with $κ_{3}= 2$, i.e., for a graph $G$ of order $n$ and size $e(G)$ with $κ_{3}(G)= 2$, we proved that $e(G)\geq (6/5)n$, and the lower bound is sharp by constructing a class of graphs, only for $n\equiv 0 \ (mod \ 5)$ and $n\neq 10$. In this paper, we improve the lower bound to $\lceil(6/5)n\rceil$. Moreover, we show that for all $n\geq 4$ but $n= 9, 10$, there always exists a graph of order $n$ with $κ_{3}= 2$ whose size attains the lower bound $\lceil(6/5)n\rceil$. Whereas for $n= 9, 10$ we give examples to show that $\lceil(6/5)n\rceil+1$ is the best possible lower bound. This gives a clear picture on the minimal size of a graph of order $n$ with generalized connectivity $κ_{3}= 2$.

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。

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英文摘要：

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