Super [formula omitted]-optimality of regular graphs

Abstract

Let G = ( V , E ) be a connected graph. An edge set S ⊂ E is a 3-restricted edge cut, if G − S is disconnected and every component of G − S has at least three vertices. The 3-restricted edge connectivity λ 3 ( G ) of G is the cardinality of a minimum 3-restricted edge cut of G . A graph G is λ 3 -connected, if 3-restricted edge cuts exist. A graph G is called λ 3 -optimal, if λ 3 ( G ) = ξ 3 ( G ) , where ξ 3 ( G ) = m i n { | [ X , X ¯ ] | : X ⊆ V , | X | = 3 , G [ X ] i s c o n n e c t e d } , [ X , X ¯ ] is the set of edges of G with one end in X 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- λ 3 for simplicity. In this paper we show that let G be a k -regular connected graph of order n ≥ 6 , if k ≥ ⌊ n / 2 ⌋ + 3 , then G is super- λ 3 .