Restricted Edge Connectivity of Harary Graphs

Abstract

An edge subset F of a connected graph G = (V,E) is a k-restricted edge cut if G − F is disconnected, and every component of G − F has at least k vertices. The k-restricted edge connectivity of G, denoted by λ _k (G), is the cardinality of a minimum k-restricted edge cut. By the current studies on λ _k , it can be seen that the larger λ _k is, the more reliable the graph is. Hence one expects λ _k to be as large as possible. A possible upper bound for λ _k is ξ _k defined as $\xi_k(G)=\min\{\omega(S):\emptyset\neq S\subset V(G), |S|=k\$and$\ G[S]\$is connected }, where ω(S) is the number of edges with one end in S and the other end in V(G) ∖ S, and G[S] is the subgraph of G induced by S. A graph G is called λ _k -optimal if λ _k (G) = ξ _k (G). A natural question is whether there exists a graph G which is λ _k -optimal for any k ≤ |V(G)|/2. In this paper, we show that except for two cases, the Harary graph has this property.

This research is supported by NSFC (10971255, 61063005), Program for New Century Excellent Talents in University (NCET-08-0921), and The Project-sponsored by SRF for ROCS, SEM.