Degree conditions for graphs to be maximally [formula omitted]-restricted edge connected and super [formula omitted]-restricted edge connected

Abstract

For a connected graph G=(V,E), an edge set S⊆E(G) is called a k-restricted edge cut of G if G−S is disconnected and every component of G−S has at least k vertices. The k-restricted edge connectivity of G, denoted by λk(G), is defined as the cardinality of a minimum k-restricted edge cut. For two disjoint vertex subsets X,Y of G, define [X,Y]={xy∈E(G):x∈X,y∈Y} and define ξk(G)=min{|[X,X¯]|:X⊆V(G),|X|=k,G[X]is connected}, where X¯=V(G)∖X. G is λk-optimal if λk(G)=ξk(G). Furthermore, G is super-λk if every minimum k-restricted edge cut of G isolates a connected subgraph with order k. The k-restricted edge connectivity is an important index to estimate the reliability of networks. In this paper, some degree conditions for graphs to be maximally k-restricted edge connected and super k-restricted edge connected are given.