Edge isoperimetric method: At least 2/3 of h-extra edge-connectivity of a kind of cube-based graphs concentrates on 2n−1

Abstract

The edge isopermetric problem on hypercube $Q_n$, proposed by Harper in 1964, is to find a vertex subset with cardinality $m$ in $Q_n$, such that the edge cut separating any vertex subset with cardinality $m$ from its complement has minimum size. Since Harper, Lindsey, Bernstein and Hart solved the edge isoperimetric problem of $Q_n$ by lexicographic order, the edge isoperimetric problem is intimately tied to many-to-many disjoint paths problem. The maximum cardinality of edge disjoint paths connecting any two disjoint connected subgraphs of order $h$ in a connected graph $G$ can be defined by the minimum modified edge-cut, called the $h$-extra edge-connectivity of $G$. It is the cardinality of the minimum set of edges in a connected graph $G$, if such a set exists, whose deletion disconnects $G$ and leaves every remaining component with at least $h$ vertices. The $(n,k)$-enhanced hypercubes $Q_{n,k}$ are constructed by adding a matching between some pair copies of $k$ dimensional subcubes $Q_k$ with $1\le k\le n-1$. The distribution of the values of the $h$-extra edge-connectivity on a recursive graph is uneven and presents a concentration phenomenon. In this paper, we start with analysing the fractal properties of the optimal solution of the edge isoperimetric problem of $Q_{n,k}$. And it is shown that although the members of $Q_{n,k}$ are not isomorphic to each other according to different $k$ where $2\le k\le n-1$, when $n$ approaches infinity, the $h$-extra edge-connectivity of $(n,k)$-enhanced hypercubes presents a concentration phenomenon. That is, for at least $2/3$ of $h\le 2^{n-1}$, the corresponding exact values ${\lambda }_h\left(Q_{n,k}\right)$ concentrate on $2^{n-1}$ for $\left(2^{n-1}/3\right)\le h\le 2^{n-1}$.