Connectivity keeping edges in graphs with large minimum degree

The old well-known result of Chartrand, Kaugars and Lick says that every k-connected graph G with minimum degree at least 3 k / 2 has a vertex v such that G − v is still k-connected. In this paper, we consider a generalization of the above result [G. Chartrand, A. Kaigars, D.R. Lick, Critically n-connected graphs, Proc. Amer. Math. Soc. 32 (1972) 63–68]. We prove the following result: e G is a k-connected graph with minimum degree at least ⌊ 3 k / 2 ⌋ + 2 . Then G has an edge e such that G − V ( e ) is still k-connected. und on the minimum degree is essentially best possible.