中文摘要：

设G是有限简单无向图，k是正整数．使G—S每个分支的阶不小于k的边割S称为G的k阶限制边割．G的四阶限制边连通度λ4（G）是G的四阶限制边割之中最少的边数．若对于任意边e∈E（G），均有λ4（G—e）=A4（G）-1，则称G是极小四阶限制边连通图．定义ξ4（G）=min{δ（U）：U包含V（G），G[U]是四阶连通导出子图}，此处δ（U）表示恰好有一个点在U上的边的数目．若λ4（G）=ξ4（G），则称G是λ4最优的．若每个5阶限制边割都孤立出G的一个5阶连通子图，则称G是超级5阶边连通的．笔者给出：极小四阶限制边连通图若不是λ4最优的，则是3正则，围长为5，任意边都关联5圈，且是超级5阶边连通的图．

英文摘要：

Let G be a finite, simple and undirected graph, and let k be a positive integer. An edge - cut S of G is called k - restricted if every component of G - S has order at least k. The minimum cardinality of all 4 - restricted edge - cuts is the 4 - restricted edge connectivity of G, denoted by λ4 （G）. A graph G is called minimally 4 - restricted edge - connected if λ4 （ G - e） =λ4 （G） - 1 for each edge e ∈ E（ G）, λ4 - optimal if λ4 （G） = ξ4 （G）, where ξ4（G） = min { δ（U） ： U belong to V（ G）, G[ U] is a connected induced subgraph of order 4 of G} ,δ（U） is the number of edges between U and V／ U, and super - λ5 if every minimum 5 - restricted edge - cut isolates a connected induced subgraph of order 5. We show that minimally 4 - restricted edge - connected graphs are λ4 - optimal except for 3 -regular graphs with girth g（G） = 5 ,each edge incident with a 5 -cycle and being super -λ5.