Restricted edge connectivity of harary graphs

• Authors: Qinghai Liu;Xiaohui Huang;Zhao Zhang
• Venue: COCOA'11 Proceedings of the 5th international conference on Combinatorial optimization and applications
• Year: 2011

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) : ø ≠ 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) = λk(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 Harary graph has this property.