Super λ3λ3-optimality of regular graphs

Let G=(V,E)G=(V,E) be a connected graph. An edge set S⊂ES⊂E is a 3-restricted edge cut, if G−SG−S is disconnected and every component of G−SG−S has at least three vertices. The 3-restricted edge connectivity λ3(G)λ3(G) of GG is the cardinality of a minimum 3-restricted edge cut of GG. A graph GG is λ3λ3-connected, if 3-restricted edge cuts exist. A graph GG is called λ3λ3-optimal, if λ3(G)=ξ3(G)λ3(G)=ξ3(G), where ξ3(G)=min{|[X,X¯]|:X⊆V,|X|=3,G[X]isconnected},[X,X¯] is the set of edges of GG with one end in XX and the other in X¯ and X¯=V−X. Furthermore, if every minimum 3-restricted edge cut is a set of edges incident to a connected subgraph induced by three vertices, then G is said to be super 3-restricted edge connected or super-λ3λ3 for simplicity. In this paper we show that let GG be a kk-regular connected graph of order n≥6n≥6, if k≥⌊n/2⌋+3k≥⌊n/2⌋+3, then GG is super-λ3λ3.