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

Let G=(V(G),E(G)) be a simple connected graph and [email protected]?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 @l^(^m^)(G) be the minimum size of all m-restricted edge cuts and @x"m(G)=min{|@w(U)|:|U|=m and G[U] is connected}, where @w(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 [email protected]^(^m^) if @l^(^m^)(G)[email protected]"m(G). An [email protected]^(^m^) graph is called super m-restricted edge-connected if every minimum m-restricted edge cut is @w(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 [email protected]^(^3^) vertex transitive graph to be super 3-restricted edge-connected. In particular, a complete characterization for an [email protected]^(^2^) minimal Cayley graph to be super 2-restricted edge-connected is obtained.

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