Conditional edge connectivity properties, reliability comparisons and transitivity of graphs

Abstract

The ith conditional edge-connectivity λ i for a (simple) graph G is the minimum cardinality of a set of edges, if any, whose deletion disconnecting G and every remaining component has more than i vertices. The usual edge connectivity λ and the restricted edge connectivity λ′ of G correspond to λ 0 and λ 1, respectively. We first give an improved reliability comparison criterion between two regular graphs by means of λ 0, λ 1 and λ 2. Next we prove that a vertex-transitive graph with degree d⩾4 and girth g⩾5 or an edge-transitive d-regular graph with degree d⩾4 and girth g⩾4 must have the maximum λ i , namely, λ i =( i+1) d−2 i for 0⩽ i⩽min( g−2, n/2−1), where n is the order of the graph. Finally, as an application of the above results, we show that both K a,a (a⩾4) and K a+1,a+1−(a+1)K 2 (a⩾5) are the most reliable graphs for sufficiently small edge failure probabilities.