On super 2-restricted and 3-restricted edge-connected vertex transitive graphs

Let G=(V(G),E(G))G=(V(G),E(G)) be a simple connected graph and F⊂E(G)F⊂E(G). An edge set FF is an mm-restricted edge cut if G−FG−F is disconnected and each component of G−FG−F contains at least mm vertices. Let λ(m)(G)λ(m)(G) be the minimum size of all mm-restricted edge cuts and ξm(G)=min{|ω(U)|:|U|=m and G[U] is connected}ξm(G)=min{|ω(U)|:|U|=m and G[U] is connected}, where ω(U)ω(U) is the set of edges with exactly one end vertex in UU and G[U]G[U] is the subgraph of GG induced by UU. A graph GG is optimal-λ(m)λ(m) if λ(m)(G)=ξm(G)λ(m)(G)=ξm(G). An optimal-λ(m)λ(m) graph is called super mm-restricted edge-connected if every minimum mm-restricted edge cut is ω(U)ω(U) for some vertex set UU with |U|=m|U|=m and G[U]G[U] being connected. In this note, we give a characterization of super 2-restricted edge-connected vertex transitive graphs and obtain a sharp sufficient condition for an optimal-λ(3)λ(3) vertex transitive graph to be super 3-restricted edge-connected. In particular, a complete characterization for an optimal-λ(2)λ(2) minimal Cayley graph to be super 2-restricted edge-connected is obtained.