Optimally restricted edge connected elementary 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 λk(G), is the cardinality of a minimum k-restricted edge cut. By the current studies on λk, it can be seen that the larger λk is, the more reliable the graph is. Hence one expects λk to be as large as possible. A possible upper bound for λk is ξk defined as ξk(G)=min{ω(S):0̸≠S⊂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 λk-optimal if λk(G)=ξk(G). A natural question is whether there exists a graph G which is λk-optimal for any k≤|V(G)|/2. In this paper, we show that except for two cases, the elementary Harary graph has this property.