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Let G be a 4-connected graph. For an edge e of G; we do the following operations on G: first, delete the edge e from G; resulting the graph G À e; second, for all the vertices x of degree 3 in G À e; delete x from G À e and then completely connect the 3 neighbors of x by a triangle. If multiple edges occur, we use single edges to replace them. The final resultant graph is denoted by G~e: If G~e is still 4-connected, then e is called a removable edge of G: In this paper we prove that every 4-connected graph of order at least six (excluding the 2-cyclic graph of order six) has at least ð4jGj þ 16Þ=7 removable edges. We also give the structural characterization of 4-connected graphs for which the lower bound is sharp. r

An edge e of a k-connected graph G is said to be a removable edge if Ge is still k-connected. A k-connected graph G is said to be a quasi (k + 1)-connected if G has no nontrivial k-separator. The existence of removable edges of 3-connected and 4-connected graphs and some properties of quasi k-connected graphs have been investigated [In this paper, we first investigate the relation between quasi connectivity and removable edges. Based on the relation, the existence of removable edges in k-connected graphs (k 5) is investigated. It is proved that a 5-connected graph has no removable edge if and only if it is isomorphic to K 6 . For a k-connected graph G such that end vertices of any edge of G have at most k − 3 common adjacent vertices, it is also proved that G has a removable edge. Consequently, a recursive construction method of 5-connected graphs is established, that is, any 5-connected graph can be obtained from K 6 by a number of + -operations. We conjecture that, if k is even, a k-connected graph G without removable edge is isomorphic to either K k+1 or the graph H k/2+1 obtained from K k+2 by removing k/2 + 1 disjoint edges, and, if k is odd, G is isomorphic to K k+1 .

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Mader conjectured that every k-critical n-connected non-complete graph G has (2k + 2) pairwise disjoint fragments. W e show that Mader's conjecture holds if the order of G is greater than (k + 2)n. From this, it implies that t w o other conjectures on k-critical n-connected graphs posed by Entringer, Slater, and Mader also hold if the cardinality of the graphs is large. 0 1995 John Wiley & Sons, Inc.