Extra edge connectivity and isoperimetric edge connectivity

Abstract

An edge set S of a connected graph G is a k-extra edge cut, if G - S is no longer connected, and each component of G - S has at least k vertices. The cardinality of a minimum k-extra edge cut, denoted by λ k ( G ) , is the k-extra edge connectivity of G. The kth isoperimetric edge connectivity γ k ( G ) is defined as γ k ( G ) = min { ω ( U ) : U ⊂ V ( G ) , | U | ⩾ k , | U ¯ | ⩾ k } , where ω ( U ) is the number of edges with one end in U and the other end in U ¯ = V ⧹ U . Write β k ( G ) = min { ω ( U ) : U ⊂ V ( G ) , | U | = k } . A graph G with γ j ( G ) = β j ( G ) ( j = 1 , … , k ) is said to be γ k -optimal. In this paper, we first prove that λ k ( G ) = γ k ( G ) if G is a regular graph with girth g ⩾ k / 2 . Then, we show that except for K 3 , 3 and K 4 , a 3-regular vertex/edge transitive graph is γ k -optimal if and only if its girth is at least k + 2 . Finally, we prove that a connected d-regular edge-transitive graph with d ⩾ 6 e k ( G ) / k is γ k -optimal, where e k ( G ) is the maximum number of edges in a subgraph of G with order k.