Super s-restricted edge-connectivity of vertex-transitive graphs

Abstract

Let G be a connected graph with vertex-set V(G) and edge-set E(G). A subset F of E(G) is an s-restricted edge-cut of G if G-F is disconnected and every component of G-F has at least s vertices. Let lambda(s)(G) be the minimum size of all s-restricted edge-cuts of G and xi(s)(G)=min{vertical bar[X, V(G)backslash X]vertical bar: vertical bar X vertical bar = s, C[X] is connected}, where [X, V(C)backslash X] is the set of edges with exactly one end in X. A graph G with an s-restricted edge-cut is called super 8-restricted edge-connected, in short super-lambda(s), if lambda(s)(C) = xi(s)(G) and every minimum s-restricted edge-cut of G isolates one component G[X] with vertical bar X vertical bar = s. It is proved in this paper that a connected vertex-transitive graph G with degree k > 5 and girth g > 5 is super-lambda(s) for any positive integer s with s <= 2g or s <= 10 if k = g = 6.