图的$k$ -边连通度的一致性和锐界
Extremality and Sharp Bounds for the $k$-edge-connectivity of Graphs
摘要：
Boesch和Chen（SIAM J. Appl. Math.，1978）提出了广义边连通度的一个截断版本，称为$k$ -边连通度。对于具有$2\leq k\leq n$的任何整数$k$，由$λ_k(G)$表示的图$G$的{ \em $k$ -边连通性}被定义为从$G$移除其产生具有至少$k$分量的图的边的最小数目。本文首先利用$n$和$k$计算了$λ_k(G)$的一些精确值和锐界。然后讨论了$λ_k(G)$与其他广义连通性之间的关系。在$\mathcal{O}(n^2)$时间将提供一个算法，使我们可以得到一个尖锐的上限方面的最大程度。在我们的结果中，我们还计算了函数$f(n,k,t)$的一些精确值和锐界，该函数定义为具有阶$n$和$λ_k(G)=t$的连通图$G$的最小尺寸。
英文摘要：
Boesch and Chen (SIAM J. Appl. Math., 1978) introduced the cut-version of the generalized edge-connectivity, named $k$-edge-connectivity. For any integer $k$ with $2\leq k\leq n$, the {\em $k$-edge-connectivity} of a graph $G$, denoted by $λ_k(G)$, is defined as the smallest number of edges whose removal from $G$ produces a graph with at least $k$ components. In this paper, we first compute some exact values and sharp bounds for $λ_k(G)$ in terms of $n$ and $k$. We then discuss the relationships between $λ_k(G)$ and other generalized connectivities. An algorithm in $\mathcal{O}(n^2)$ time will be provided such that we can get a sharp upper bound in terms of the maximum degree. Among our results, we also compute some exact values and sharp bounds for the function $f(n,k,t)$ which is defined as the minimum size of a connected graph $G$ with order $n$ and $λ_k(G)=t$.
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