Contractible edges and triangles in k-connected graphs

It is proved that if G is a k-connected graph which does not contain K-4(-), then G has an edge a or a triangle T such that the graph obtained from G by connecting e or by contracting T is still k-connected. By using this theorem, we prove some theorems which are generalizations of earlier work. In addition, we give a condition for a k-connected graph to have a k-contractible edge, which implies two theorems proved by C. Thomassen (1981, J. Graph Theory 5, 351-354) and by the author (2001, Australas. J. Combin. 24, 165-168). respectively. (C) 2002 Elsevier Science (USA).