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

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